*[Chapter 5.1*] 007 .ta 56r 006 .tg NI 005 .ul 1 059 Chapter 5 : Refinement of Approximate Molecular Structures. 005 .tc : 030 :"You've never had it so good" 010 :Macmillan 003 .tc 006 .tg NA 050 Basic thermodynamics shows that the degree to 063 which a given reaction will progress depends on the free energy 059 change in that reaction; the more negative the free energy 060 change, the more that reaction will tend to completion. In 060 terms of the structure of a molecular complex, the lower the 060 free energy of a given configuration, the more probable that 061 configuration is. For reasons discussed in chapter 2, it is 072 assumed that the internal energy of the complex *Iin vacuo *Nis the most 059 important term in producing this free energy and that other 050 terms, whilst perhaps not paralleling the internal 047 energy, do not outweigh it. Likely structures 064 for a complex therefore correspond to low energy configurations. 011 The sharper 064 and deeper a minimum is, the more confident we can feel that the 058 structure will correspond closely to that found in nature, 019 provided that there 064 is some reaction pathway connecting the uncomplexed constituents 061 with the final complex structure. In this chapter, we study 065 ways of finding these minimum energy configurations automatically 065 using the semi-empirical formulations of interaction energy given 011 previously. 006 .tg NI 005 .ul 1 041 5.1 : Methods used for Optimisation*S1*N. 006 .tg NA 069 We used two basic methods for locating minimum energy structures 054 of complexes and these can be described most simply as 067 "gridsearch" and "gradient" techniques respectively. Suppose, for 066 described in terms of "n" variables (and we are unconcerned at the 054 moment what these variables actually are), then we may 060 consider the variables as forming an n-dimensional space and 064 the problem is to find the minimum value of some energy function 080 f(x*d1*l,...,x*dn*l) within that space. Gridsearch techniques construct a mesh 067 of points over the space and evaluate the function at those points, 064 and it is immediately obvious which points correspond to minima. 059 Gradient techniques, on the other hand, start off from some 059 arbitrary point in the space and use the function value and 063 its corresponding gradient vector to move to successively lower 060 positions, until a minimum is located to within the required 063 tolerance. Both techniques have advantages and disadvantages. 064 A gridsearch technique is inherently very simple. All one 065 needs to do is produce a set of configurations covering the range 063 of values possible for the different variables and evaluate the 064 energy of each. Unfortunately, it is obvious that this becomes 061 impossibly time-consuming even with very few variables, since 062 the number of points which must be sampled grows exponentially 029 with the number of variables. 054 This limits the utility of the method to systems which 057 can be described in terms of a small number of variables. 062 On the other hand, one can see immediately from the results of 061 a gridsearch calculation what minima exist, whether there are 069 low-energy pathways between them and the starting point, and what the 052 rest of the conformational space looks like. There 064 are no problems associated with convergence such as we shall see 056 with gradient techniques. In other words, one can feel 050 and about the validity of the results it produces. 066 Gradient routines of the type we shall discuss here perform a 065 series of "linear-searches". Given an arbitrary starting point, 063 the algorithms determine a direction in the n-dimensional space 068 and the minimum function value is located in that direction. A new 063 search direction is then chosen and the entire process repeated 060 until some termination condition is satisfied. The various 058 gradient algorithms use different methods for choosing the 073 succession of search directions and we shall discuss three of these here. 059 Conceptually, the simplest method is that known as the 064 "Steepest Descent" algorithm. Each new line search takes place 062 along the direction in which the gradient is steepest, so that 052 the succession of points *Dx_*I*dk *N*lgenerated is: 006 .sp 1c 005 .ce 1 061 *Dx_*I*dk+1 *N*l= *Dx_*I*dk *N*l- *S*(*I*t***dk*)*D*lg_*I*dk 010 *N*l.sp 1c 099 where *Dg_*I*dk *N*lis the gradient vector and the scalar *S*(*I*t***dk*) *N*lindicates the use of 070 a linear-search within the routine. This would appear to ensure that 071 the function value always decreases during a search, which is true, but 026 the rate of convergence of 052 the algorithm can become very slow as the minimum is 064 approached. This is because the method takes no account of the 054 variation of curvature of the function with direction, 032 information which is provided by 042 the second derivative, or Hessian, matrix. 064 The other algorithms which we shall discuss generate search 005 .ne 4 064 directions which are conjugate*t** *lto one another with respect 005 .fn 1 032 *GG *Nif *Dp_*tT*G*lG*Dq_ *N= 0. 005 .en 1 067 to this Hessian matrix. It can be shown, for a quadratic function 059 (that is, one involving only terms of order two or less) of 064 n variables, that the minimum can be located in n searches along 062 conjugate directions if each search is performed exactly. In 052 other words, for quadratic functions, convergence is 066 guaranteed for these algorithms, which is obviously an improvement 042 on the steepest descent method. In fact, 069 convergence for non-quadratic functions is also improved considerably 035 by the use of conjugate directions. 073 Analytically, the succession of points *Dx_*I*dk *N*lwhich should be 085 generated for a quadratic function f(*Dx_*N) = *A*Dx_*tT*G*lG*Dx_ *N+ *Dg_x_ *N+ *Ic 049 *Nin terms of the Hessian matrix *GG*I*dk *N*lis: 006 .sp 1c 005 .ce 1 067 *Dx_*I*dk+1 *N*l= *Dx_*I*dk *N*l- *GG*(*N*t-*I*dk*)*N*t1*D*lg_*I*dk 010 *N*l.sp 1c 041 For a more general function, we would use 006 .sp 1c 005 .ce 1 085 *Dx_*I*dk+1 *N*l= *Dx_*I*dk *N*l- *S*(*I*t***dk*)*G*lG*(*N*t-*I*dk*)*N*t1*D*lg_*I*dk 010 *N*l.sp 1c 083 where the *S*(*I*t***dk*) *N*lindicates the need to locate the minimum along lines 010 of search. 071 However, the evaluation of the matrix *GG*I*dk *N*land its inverse 086 *GG*(*N*t-*I*dk*)*N*t1 *lcan be very time-consuming, and can also require considerable 061 storage space if there are many variables. When the storage 056 space is not a problem, the class of algorithms known as 065 "quasi-Newton" can give good results. These algorithms build up 093 an approximation *GH*I*dk *N*lto *GG*(*N*t-*I*dk*)*N*t1*l, which is initially the unit matrix 057 (making the first search a "steepest descent" search) but 042 the minimum is approached. The method of 070 updating *GH*I*dk *N*lvaries with the precise algorithm used; when we 064 used this type of algorithm, we updated *GH*I*dk *N*lby means of 054 the complementary Davidon-Fletcher-Powell*S2 *Nmethod. 062 If storage space is a problem, then a useful algorithm is 068 the conjugate gradient method suggested by Fletcher and Reeves*S3*N, 061 which generates the successive search directions using vector 032 arithmetic only. We have here: 003 .nj 003 .nf 006 .sp 1c 074 *Dx_*I*dk+1 *N*l= *Dx_*I*dk *N*l+ *S*(*I*t***dk*)*D*lp_*I*dk 073 *N*lwhere *Dp_*I*dk *N*l= -*Dg_*I*dk *N*l+ *Sb*I*dk*D*lp_*I*dk-1 059 *N*land *Sb*I*dk *N*l= 0, *Ik *N= 1 077 = *S?*Dg_*I*dk*S*l? *N/ *S?*Dg_*I*dk-1*S*l?*N, *Ik *S> *N1 003 .ju 003 .fi 006 .sp 1c 055 The first step is once again a steepest-descent search. 072 Results by Huang and Levy*S4*N, and later by Dixon*S5*N, have shown 053 that, when the minima along lines of search are found 064 exactly, then from a given starting point, the search directions 062 generated by various conjugate direction algorithms, including 060 complementary DFP and Fletcher-Reeves, are identical. This 058 would suggest that there was little point in incurring the 056 overhead of matrix arithmetic required with quasi-Newton 065 algorithms. However, this result holds only if the exact minima 066 are found along lines of search, and in practice this condition is 062 not satisfied. Usually, quasi-Newton algorithms are found to 064 approach minima more quickly than conjugate gradient algorithms. 064 This brings us to the next point, which concerns the method 065 to be of less importance than how these directions are generated. 066 In fact, there is considerable scope for wasting much machine-time 053 with the functions of interest here, if inappropriate 039 methods are used for the line-searches. 061 Conventional line-search routines follow the type of strategy 012 given below: 006 .sp 1c 006 .ll -3 006 .in 10 006 .ti -5 076 1.ZZZDetermine g1 = *Dg_*N.*Dp_*N, the gradient along the line of search; if 021 g1 *S> *N0 then exit. 006 .ti -5 068 2.ZZZ(Initial step) Take some step, a, and evaluate g2 = *Dg_*N.*Dp_ 022 *Nat the new position. 006 .ti -5 075 3.ZZZ(Extrapolation) If g2 *S< *N0, set a = ka (where k is commonly 2), set 031 g1 = g2, and go back to step 2. 006 .ti -5 070 4.ZZZ(Interpolation) Otherwise, minimum along line has been bracketed. 070 Fit some curve (commonly a cubic or a parabola) between the two points 065 bracketing the minimum and move to the (algebraically-determined) 022 minimum of this curve. 006 .ti -5 064 5.ZZZReplace one of the two points used in (4) by this new point 066 and repeat operations (4) and (5) until some termination criterion 007 is met. 006 .ll +3 005 .in 0 067 The termination condition mentioned in (5) is usually that the 065 size of the gradient along the line of search has been reduced to 054 some proportion of its value at the start of the line. 059 The smaller this factor, the more accurately the minimum is 067 located along the line of search, but the more function evaluations 053 will be needed to locate it. It has been shown that 069 algorithms using conjugate directions converge satisfactorily if this 076 search should be reduced to half its initial value (Wolfe's condition*S6*N). 002 *? *[Chapter 5.2*] 072 Haigh*S7 *Nhas investigated the behaviour of linesearch algorithms, 062 particularly when applied to functions of the type we consider 058 here, and has noted five main places in which conventional 026 algorithms are inadequate: 006 .sp 1c 006 .in 10 006 .ll -3 006 .ti -5 062 1.ZZZIn the choice of the initial step length. Some existing 058 optimisation routines are supplied with an estimate of the 062 minimum to be located. They then assume that the function is 066 parabolic, with given minimum and a known gradient at the start of 060 the line, and estimate an initial step-length on this basis. 043 Apart from rather begging the question, two 019 problems can arise: 006 .in +3 006 .ti -3 059 a.ZIf the estimate of the minimum is too high, the function 064 value at the start of the line will eventually become lower than 059 this estimate, which means that the equation specifying the 044 initial step will have no sensible solution. 006 .ti -3 064 b.ZIf the estimate is too low, then the initial step will become 047 excessively large as the minimum is approached. 006 .in -3 003 .br 063 Haigh solved this problem by basing the initial step-length for 058 linesearches after the first on estimates of the curvature 022 from earlier searches. 006 .ti -5 072 2.ZZZIn the method of extrapolation. Traditionally, each extrapolation 068 along the line of search doubled the previous step. If the initial 069 step was wrong by several orders of magnitude (as it quite frequently 060 was), then this would mean performing an excessive number of 069 function was parabolic and to increase the step length on this basis, 043 taking special measures if the curvature of 038 the function was found to be negative. 006 .ti -5 067 3.ZZZIn the method of interpolation. If the initial step made was 066 too large, then one of the two points bracketing the minimum would 051 be much further away from the minimum in value than 056 the other, and a phenomenon which he called "tortoising" 067 would occur. When a parabola was fitted between these two points, 064 the minimum calculated would be very close to the lower of them. 047 The process of fitting a parabola would then be 067 repeated a very large number of times before the true minimum along 062 the line of search was located. This was overcome by testing 045 for the situation on the first extrapolation, 042 and, if it occurred, backtracking from the 052 point with high function value before interpolating. 006 .ti -5 058 4.ZZZIn the method used to decide termination of a search. 057 We have mentioned Wolfe's criterion that convergence of a 062 conjugate direction algorithm is ensured if the gradient along 067 each line of search is reduced by at least half. Haigh found that 055 it was inefficient to demand higher accuracy than this. 006 .ti -5 060 5.ZZZBecause of the possibility of jumping out of the valley 063 containing a low minimum into another containing a higher local 059 minimum. The conventional logic would interpolate towards 063 a minimum whenever it found two successive points with opposite 068 gradients along the line of search, without checking whether in fact 064 these points bracketed a minimum with a value higher than points 068 elsewhere on the line. This problem was overcome by taking greater 053 care in the logic used to test for bracketing minima. 005 .in 0 064 Haigh had developed these conclusions mainly from examining 060 the behaviour of optimisation algorithms on certain standard 058 mathematical problems and also by studying small molecular 062 systems in which individual atoms interacted electrostatically 056 and through van der Waals potentials. Typical of these 063 was a system containing two O-H dipoles positioned close to one 059 another. After making the improvements mentioned above to 067 the standard linesearch algorithm, he found that the minimum energy 064 configuration of this system could be located in about one third 062 of the number of function evaluations required by the standard 011 algorithms. 061 The use of semi-empirical formulae of the type described 068 in this thesis to calculate energies of molecular systems results in 050 an overall function with the following properties: 006 .sp 1c 006 .ll -3 006 .in 10 006 .ti -5 066 1.ZZZIt does not behave quadratically except close to the minimum. 006 .ti -5 071 2.ZZZSome changes in variables (for example those which move non-bonded 065 atoms further apart) result in relatively small changes in energy 053 whereas others (those which stretch chemical bonds or 066 which move non-bonded atoms too close to one another) produce very 030 much larger changes in energy. 006 .ll +3 005 .in 0 059 When these functions are applied to systems containing 062 several hundred atoms (as they were here), two other important 018 properties appear: 006 .sp 1c 006 .in 10 006 .ll -3 006 .ti -5 005 .ne 5 058 3.ZZZEvaluation of the energy function is expensive*t***l. 005 .fn 1 061 of this function with carefully optimised code took about ten 008 seconds. 005 .en 1 006 .ti -5 060 4.ZZZRounding errors incurred during the function evaluation 022 become non-negligible. 006 .ll +3 005 .in 0 060 Points 1 to 3 provided good indication that attempts to 062 optimise molecular structures using semi-empirical interaction 064 potentials would be fraught with difficulty, and consequently we 068 used Haigh's linesearch algorithm throughout our studies (making one 057 or two minor improvements which will be mentioned later). 062 We did, however, perform one or two checks on the behaviour of 067 standard library algorithms such as those provided by NAG*S8*N; we 062 found that these algorithms behaved erratically, using grossly 055 excessive amounts of computer-time, when used to refine 044 structures containing several hundred atoms. 061 Point 4 warrants further discussion. When evalation of 067 a function involves calculating terms, each of which is obtained to 059 a certain precision, the relative error tends to accumulate 025 as the square root of the 065 number of terms. For example, if individual atom-atom terms are 071 obtained to ten significant figures (as would be the case on a computer 067 like the ICL 1906A) and there are ten thousand terms to accumulate, 068 the final answer will probably hold about eight significant figures. 066 Whenever an optimisation algorithm is taking steps small enough so 067 that the difference in energy between two successive points is less 056 than one part in 10*S8*N, the calculated difference will 066 be entirely without meaning. It is necessary to make appropriate 060 checks in optimisation algorithms to ensure that termination 040 occurs before this situation is reached. 067 Whilst points which differ in energy by only one part in 10*S8 067 *Nwill be so close together that it is physically not worthwhile to 070 procede further with the refinement, it is quite possible to calculate 072 the interaction terms in a way which gives far fewer significant figures 063 in the answer. We found that many of the approximations which 073 have commonly been used to speed up calculation of individual interatomic 069 terms produced effects which were akin to large-scale rounding error. 015 These included: 006 .ll -3 006 .in 10 006 .ti -5 006 .sp 1c 064 1.ZZZUse of "table look-up" techniques, in which the interaction 069 potentials for given atomic separations are obtained by interpolating 030 between two values in a table. 006 .ti -5 077 2.ZZZUse of polynomial approximations to functions such as *De*I*tx*N*l, when 044 only a few terms are included in the series. 006 .ti -5 071 3.ZZZNot evaluating interactions between atoms separated by more than a 017 certain distance. 006 .ti -5 077 4.ZZZUse of a function with a discontinuous derivative (such as the piecewise 070 linear approximation to a dielectric constant described in chapter 2). 006 .ll +3 005 .in 0 073 As an example of what can happen if these approximations are used to 069 speed up function evaluation, in figure 5.1 we show two contour plots 064 of the interaction energy between a dinucleotide molecule and an 056 intercalated acridine molecule, as the acridine is moved 074 *Ien masse *Nin its site in the dinucleotide. In one case the individual 068 interatomic terms were calculated to the full precision available on 075 the machine, whereas in the other case, approximations 1,3 and 4 were made. 071 being moved by about 1A in directions perpendicular to the helix axis.) 073 Whilst the two minimum energies found are almost identical, it is obvious 068 that an optimisation algorithm would locate one minimum considerably 061 faster than the other. What had been gained in the speed of 049 evaluation of the energy of any configation would 063 be lost several times over in the greater number of evaluations 031 required to locate the minimum. 052 Having learned the importance of avoiding those 065 "short-cuts" described above when evaluating the energy function, 067 we found that both the Fletcher-Reeves conjugate gradient algorithm 064 and the complementary DFP quasi-Newton algorithm were capable of 063 refining rapidly molecular structures possessing few degrees of 065 freedom, when coupled with Haigh's linesearch routine. We found 064 one main shortcoming in these circumstances, which was concerned 054 with the phenomenon of "tortoising". Haigh's routine 063 had tested whether the initial step made along a line of search 063 produced a new point on the opposite side of the minimum to the 058 starting point which had a much higher function value than 067 the starting point; if this occurred, then the routine backtracked 067 from the offending point. We found that the situation could arise 034 at any stage in the extrapolation, 070 owing to the rapid rise in non-bonded potentials at separations closer 066 than van der Waals minima. This fault was cured by ensuring that 063 points which bracketeed a minimum were of comparable magnitude, 020 and the behaviour of 056 the optimisation algorithms was improved in consequence. 063 Table 5.1 shows how the modified algorithms performed when 065 in fact was the system (9-aminoacridine.H+:dApA)). The variables 069 of the refinement were three coordinates describing the bulk position 070 of the acridine, which was therefore constrained to move *Ien masse*N. 063 With these degrees of freedom, the energy function behaves well 068 (see figure 5.1(a) for a two-dimensional slice through this function 058 for a similar system) and rapid convergence to the minimum 068 should be possible. As can be seen from the table, both algorithms 063 did locate the minimum rapidly; the conjugate direction method 058 required sixteen function evaluations to locate it to four 066 significant figures, whereas the quasi-Newton method required ten. 069 The quasi-Newton method is seen to be slightly better in terms of the 062 number of linear searches required to attain a given accuracy, 061 but is about twice as good in terms of the number of function 065 evaluations required. This is due to later linesearches made by 065 the quasi-Newton algorithm requiring only one function evaluation 058 to locate a minimum, because the algorithm has built up an 045 accurate approximation to the Hessian matrix. 062 The remainder of this chapter is devoted to a description 060 of the studies we made of intercalation complexes which were 062 based on numerical refinement of an energy function. Many of 044 the results can only be described by quoting 045 values obtained for interaction energies, and 065 we must point out here that these values depend critically on the 063 molecular fragments used in the calculations. This is largely 063 because of the ionic nature of the individual molecules and the 056 long range of electrostatic interactions. (The problem 065 constants for crystal structures*S9*N.) Consequently, it is not 067 possible to draw conclusions on the basis of the *Iabsolute *Nvalue 063 of the interaction energies quoted, other than that they appear 060 to be of the right order of magnitude. Comparisons between 068 *Irelative *Nvalues are more valid; we have tried wherever possible 039 to ascertain that the conclusions drawn 061 are unaffected by the precise molecular fragments used in the 013 calculations. 064 We start by studying the stacking of acridines in solution, 065 then examine the interactions of acridines with nucleic acid, and 057 finally see how daunomycin intercalation relates to this. 002 *? *[Chapter 5.3*] 006 .tg NI 005 .ul 1 049 5.2 : Stacking of Acridine Molecules in Solution. 006 .tg NA 048 Some acridines aggregate in ionic solutions 034 by stacking on one another and the 059 degree of association varies depending on the nature of the 070 solvent, its ionic strength and on which member of the acridine family 076 is being studied. For example, Robinson and co-workers*S10*N, studying the 073 behaviour of acridine orange (3,6-di-(dimethylamino)-acridinium chloride) 065 in solution found that the equilibrium constant for the reaction: 006 .sp 1c 005 .ce 1 025 2A -*S> *NA*S...*NA dimer 006 .sp 1c 069 increased if the ionic strength of the solution increased and also if 069 small quantities of methanol, dioxan or urea were added. They found 084 that, in 0.1M NaCl, the free energy change *SD*DG*(?*T1*)2 *Nwas about -24kJZmol*t-1 066 *land that the enthalpy of the reaction was about -38kJZmol*t-1*l. 069 An approximate analysis of the various contributions to this enthalpy 065 acridine molecules and that other terms were small by comparison. 070 An intuitive model for the acridine stacks would place successive 051 molecules in van der Waals contact with one another 037 (and separations of between 3A and 4A 075 are suggested spectroscopically*S11*N), with the molecules aligned parallel 055 to one another so as maximise overlap. The similarity 075 between this model and the conventional model for intercalation is obvious; 071 instead of having a continuous stack of acridines, intercalation merely 036 replaces most of them by base-pairs. 075 However, there are two points to be noted before we accept this rather 071 nice model of acridine stacking. Firstly, Wright*S11 *Nhas found that 070 3,6-di-(t-butylamino)-acridine molecules stack with similar degrees of 064 association to other acridines. It is manifestly impossible to 065 align these t-butyl compounds parallel to one another at the same 067 spacing that would be found with a more completely planar acridine, 068 since the bulky t-butyl groups would interact very unfavourably with 070 one another. Therefore, it must be possible to introduce some degree 063 of randomness into the orientation of the stacked molecules (by 069 rotating each one relative to the one above, whilst still maintaining 064 the stacking) without losing much interaction energy. At first 069 sight, this would appear to be at variance with Robinson's conclusion 074 that the majority of the interaction energy results from dispersion terms, 039 since introducing such randomness would 043 lose much of the overlap between molecules. 071 More importantly, the individual acridine molecules are protonated 066 under the conditions of the experiments carried out to demonstrate 072 positive charges 3.5A apart, and although this would be reduced somewhat 077 when the charge was delocalised over the acridine molecules, our calculations 068 show that there is still a residual electrostatic interaction energy 054 of some 160kJZmol*t-1 *lbetween acridinium ions placed 072 3.5A apart in vacuo. This has great bearing on the neighbour-exclusion 060 principle of intercalation; if the protonated acridines can 064 be stable stacked 3.5A apart in ionic solution when an empirical 063 calculation would show them to be very much destabilised, it is 055 unreasonable to suppose that repulsion between residual 059 charges on these acridines prevents them from intercalating 020 at sites 6.8A apart. 068 Quite clearly, salt counterions in solution must be aggregating 068 around the acridine stacks so as to reduce the overall electrostatic 069 contribution to the interaction energy. This is confirmed partly by 073 the dependence of the degree of association on ionic strength, and partly 061 by the fact that the entropy of reaction becomes increasingly 072 unfavourable as the acridine concentration increases, demonstrating that 073 an increasing amount of ordering is taking place to stabilise the stacks. 070 Robinson regards this aggregation as a dielectric effect; we consider 068 this to be a rather loose use of the term and also not very helpful, 063 for we have no way of relating the ionic strength of a solution 060 around the system to the dielectric constant which should be 066 included when evaluating the interactions betwen residual charges. 070 We considered that a reasonable model for acridine stacking would 067 place successive protonated acridines in van der Waals contact with 070 with negative salt ions aggregating in the groove between each pair of 069 acridines so as to reduce the electrostatic interaction between them. 062 It was not possible to perform rigorous calculations on such a 063 model treating it as a set of discrete components, owing to the 059 large number of salt ions and solvent molecules which would 063 have had to be included. We therefore considered various less 062 rigorous techniques which might provide some information about 028 the properties of the model. 060 Firstly, we considered treating the aggregation of salt 060 ions around the acridine stack by means of a continuous band 063 possessing net negative charge close to the stack but gradually 054 merging into a continuous polarisable and electrically 063 neutral dielectric. This could be treated as a boundary value 063 problem; inside the cylinder defined by the stacked acridines, 061 the electrostatic potential would satisfy Laplace's equation: 006 .sp 1c 005 .ce 1 014 *SV2y*T1 *N= 0 006 .sp 1c 067 whereas, outside this region, the potential would satisfy Poisson's 005 .ne 3 009 equation: 006 .sp 1c 005 .ce 1 025 *SV2y*T2 *N= *Ik*T2*Sy*T2 008 *N.sp 1c 068 (where *Ik*T2 *Nis a screening constant, obtained, for example, from 067 Debye-Huckel theory) and the usual boundary conditions would apply, 061 that the potential and the normal component of its derivative 040 must be continuous across the interface. 067 In fact, this model is very similar to one used by Gilbert and 074 Claverie*S12 *Nto determine the effects of ionic strength on the degree of 063 intercalation of acridines into DNA, the only major differences 069 being that they included a fixed band of negatively-charged phosphate 042 acridine molecules were spaced 6.8A apart. 061 Their model of intercalation was stabilised by the attractive 066 electrostatic interactions between positive acridines and negative 059 phosphate groups and the solution to the equations defining 062 the system (similar to those given above, with another Laplace 065 equation describing the phosphate band) showed quite convincingly 048 that the size of this *Ifavourable *Ninteraction 047 was reduced as the ionic strength increased, in 070 accordance with experimental results. We felt that the model we have 061 described for acridine stacking was so similar to their model 063 that the same general conclusions would apply; the size of the 069 *Iunfavourable *Ninteraction between acridinium ions would be reduced 062 as the ionic strength increased. Correspondingly, we did not 032 pursue this calculation further. 064 It is of interest to enquire what excess negative charge is 059 required around a pair of stacked acridine ions in order to 067 counteract their electrostatic repulsion. A simple model was used 066 to answer this question; a pair of acridine ions were arranged in 069 a dimer separated by 3.4A (the normal van der Waals contact distance) 065 and we generated a series of configurations in which two chloride 069 ions of given charge were placed in the groove between the acridines, 054 determining the internal energy of the system for each 051 configuration, calculated as a sum of electrostatic 028 and dispersion energy terms. 065 Figure 5.2 shows the variation in the minimum energy of this 075 system as a function of ionic charge, for three separations of the acridine 063 ions. It can be seen quite clearly that only relatively small 067 whilst it would be foolish to read too much into the precise values 058 obtained, a residual charge of about -0.3 electronic units 067 yields an interaction energy for the system which is similar to the 071 enthalpy measured experimentally*S10*N. The reason why a low negative 071 charge can stabilise the repulsion between two unit positive charges is 052 that these positive charges are delocalised over the 071 entire acridine molecule; a charge of -0.3 units on an ion is as great 059 as the residual charge on any of the atoms in the acridine. 073 We calculated the interaction between two stacked acridine molecules 067 arising from dispersion energy terms, as a function of the relative 048 orientation of the two acridines. Robinson*S10 078 *Nestimated that this dispersion energy amounted to some -40kJZmol*t-1*l, but, 041 along with other workers, considered that 043 the stacked acridines were aligned parallel 073 to one another. We set up this model initially, with various members of 073 the acridine family, and calculated the dispersion energy as one molecule 067 was rotated about an axis passing through the centre of both. The 059 variation found for two 10-methylacridinium cations stacked 075 3.36A apart is shown in figure 5.3; 0*D? *Non this plot corresponds to the 073 two molecules being superimposed, atom for atom, on one another. It can 077 be seen that, apart from the region around 0*D?*N, where the methyl hydrogens 075 are placed unfavourably close to one another, the dispersion energy remains 070 remarkably constant at around -20kJZmol*t-1 *lfor all rotation angles. 070 With separations of around 3.25A between molecules, the minimum energy 061 was found to become somewhat lower, and with other members of 062 comparable to the value suggested by Robinson from analysis of 021 experimental results. 063 The following conclusions can be drawn from these results: 006 .in 10 006 .ll -3 006 .ti -5 006 .sp 1c 069 1.ZZZThat large interaction energies arising from electrostatic terms 049 must be treated with great caution if the systems 040 are in an ionic solution, because little 049 aggregation of positive and negative charges need 071 occur before the electrostatic interactions are completely neutralised. 044 (However, electrostatic interactions between 074 residual charges far removed from solvent are not necessarily negligible.) 006 .ti -5 006 .sp 1c 075 2.ZZZThat dispersion forces between large planar molecules are non-specific 073 and the subtle interplay between the interactions of closely spaced atoms 074 and those of others further apart means that the total interaction between 069 two planar molecules stacked on one another can be fairly independent 040 of the orientation of the two molecules. 070 (A similar conclusion has been obtained by Polozov*S13 *Nin connection 070 with the stacking energies of nucleic acid base-pairs on one another.) 006 .ti -5 006 .sp 1c 068 3.ZZZThat Scheraga's*S14 *Nparameterisation of dispersion potentials 037 predicts optimum interaction energies 066 between aromatic heterocycles which are similar to those suggested 059 experimentally. However, the predicted optimum separation 055 of stacked molecules is some 3-5*A> *Nlower than values 030 obtained crystallographically. 005 .in 0 002 *? *[Chapter 5.4.1*] 005 .ul 1 006 .tg NI 029 5.3 : Acridine Intercalation. 006 .tg NA 061 between dinucleotides and intercalated acridines, in order to 052 explain the discrepancies arising in the correlation 068 between the acridines' degree of protonation and their antibacterial 071 properties (see figure 1.5). The calculations were made using methods 061 which we have since found to exaggerate interaction energies. 020 However, checks made 061 using CNDO/2 residual charges and Scheraga's parameterisation 056 of van der Waals terms showed that the following general 028 conclusions were reasonable: 006 .sp 1c 006 .ll -3 006 .in 10 006 .ti -5 061 1.ZZZThat the interaction between a protonated acridine and a 065 dinucleotide was appreciably greater than the corresponding value 030 for the unprotonated compound. 006 .ti -5 064 2.ZZZThat the interaction energy for the protonated acridine was 065 sufficiently high to result in the formation of a stable complex, 055 taking into account the likely loss in solvation energy 017 on intercalation. 006 .ti -5 066 3.ZZZThat the interaction energy between unprotonated acridine and 068 nucleic acid was generally insufficient to produce a stable complex. 061 However, in the case of acridines possessing a nitro-group, a 070 relatively large electrostatic interaction was found with both neutral 054 and protonated forms, and such compounds were expected 051 to interact well with nucleic acid in either state. 006 .ti -5 072 4.ZZZThat high interactions between protonated acridine and nucleic acid 066 could be obtained both with the long axis of the acridine parallel 058 to the base-pairs (Lerman's model*S15*N) and also with the 070 long axis at right angles to this (Pritchard's model*S16*N), but that, 063 slightly higher values were found with Pritchard-type complexes 022 than with Lerman-type. 006 .ti -5 067 5.ZZZThat compounds containing substituent groups which led to high 050 charge separations interacted more favourably with 066 dinucleotides containing G-C base-pairs than with those containing 010 A-T pairs. 006 .ll +3 005 .in 0 062 We now propose to give some explanation for these results 058 and also to study other aspects of acridine intercalation. 065 Many of the attempts which have been made to rationalise the 066 different properties of intercalation complexes have been based on 065 two assumptions: firstly, that protonation of an acridine places 063 a unit positive charge on the ring nitrogen, and secondly, that 064 suitably placed amino-groups can hydrogen-bond to DNA phosphate. 056 In figure 5.4, we indicate residual charges found on the 072 atoms in protonated acridine, 9-aminoacridine and proflavine, calculated 005 .ne 5 071 using the CNDO/2 approximation*t***l. (Charges on hydrogen atoms have 005 .fn 1 070 *t** *lCalculations of the residual atomic charges in acridine cations 065 have been made before, for example by Sharpless and Greenblat*S17 082 *Nand by Lang and L*(*S?*No*)ber*S18*N. It seems unfortunate that the sum of the 065 residual charges published by these workers was in no case unity. 005 .en 1 060 been condensed onto the heavy atom to which they are bound.) 059 Note that, whilst formally protonated on the ring nitrogen, 063 in all cases, very little residual charge is left on this atom, 061 which means that one of the reasons put forward by Pritchard, 069 Blake and Peacocke*S16 *Nfor their model of intercalation, namely the 061 ability of the charge on ring nitrogen to interact favourably 066 Substituting the acridines at positions 3 and 6 with amino-groups, 045 giving proflavine, makes little difference to 062 the amount of charge on the central ring (slightly over half), 048 but does result in almost one third of a unit of 044 charge moving onto each of the amino-groups, 053 leaving the remainder of the atoms in the outer rings 060 more negatively charged than in acridine. The calculations 060 described in chapter 3, showed that an *Ioutwards *Nrotation 053 of the phosphate groups was likely on helix extension 063 and this would disfavour hydrogen-bonding with the amino-groups 054 in compounds like proflavine. However, electrostatic 068 interactions are of longer range than hydrogen-bonding interactions, 067 and a movement of positive charge outwards onto the amino-groups in 059 proflavine would tend to stabilise the complex, even though 074 hydrogen-bonding could not occur. Previous results*S19 *Nhave shown that 061 hydrogen-bonding was not necessarily important in stabilising 058 complexes between ethidium analogues and nucleic acid, and 063 we would suggest that the changes in static charge distribution 050 caused by substituents on the aromatic nucleus are 015 more important. 068 To determine how the interaction energy between an intercalated 059 acridine and nucleic acid varied as a function of unwinding 074 angle at the intercalation site, we set up a model of (proflavine.H*t+ *l: 041 d(A-A).d(T-T)) in a Lerman-type structure 025 using dinucleotide models 080 with *SD*DB *Nvalues between 3*D? *Nand 39*D?*N, and refined the position of the 066 intercalated acridine by means of the quasi-Newton algorithm*S2*N. 066 The energy, calculated as a sum of van der Waals and electrostatic 073 norm of its gradient was typically less than 0.0001 kJ mol*t-1 *lA*t-1*l. 073 The variation in energy as a function of *SD*DB *Nis shown in figure 5.5, 066 and is seen to be relatively small over the range 15*D?*N-25*D?*N, 070 with a minimum when the dinucleotide was unwound by about 13*D? *Nfrom 063 the normal B-DNA value. Interestingly, this is the same value 063 which resulted from the process of helix extension described in 066 chapter 3. Measurements of supercoil reversal by proflavine have 080 suggested*S20 21 *Nthat 18*D? *Nunwinding occurs on binding to DNA, which is not 057 dissimilar to this figure. The minimum energy structure 057 is illustrated in figure 5.6; as well as a space-filling 064 representation looking towards the wide groove, we also show the 062 overlap between acridine and base-pairs looking down the helix 055 axis, and compare it with structures suggested by Patel 073 and Canuel*S22*N, from nmr data, and by Alden and Arnott23, from computer 064 modelling calculations. Although we used a different base-pair 061 sequence from these workers (ApA instead of CpG), it is clear 061 that the overlap between acridine and base-pairs found in our 070 calculations corresponds more closely to the experimentally-determined 044 structure than to the theoretical structure. 002 *? *[Chapter 5.4.2*] 032 The position of the minimum 060 was determined by an interplay between various van der Waals 066 terms; starting from a structure unwound by 36*D?*N, in which the 055 base-pairs were superimposed on one another, increasing 076 *SD*DB *Nproduced a slight increase in the van der Waals interaction between 056 base-pairs and the acridine. Further winding swung the 062 between positive acridine and negative phosphate could only be 065 achieved by bringing the acridine into unfavourably close contact 065 with these sugar groups. It is interesting to see how the total 065 interaction energy was made up; for the minimum energy structure 065 (*SD*DB *N= 23*D?*N), a breakdown of terms is given in table 5.2. 071 Note the dominance of the electrostatic terms; favourable interactions 066 with the phosphate groups and the bases are offset by unfavourable 060 interactions with the sugar groups. (Interactions with the 055 phosphate groups are favourable, largely because of the 067 formal negative charge on these groups, but also because the oxygen 072 atoms linking phosphorus to sugar have abstracted some additional charge 064 from the sugar ring. A similar abstraction of about -0.2 units 065 of charge by each of the base groups makes these bases negatively 068 charged, but leaves a residual positive charge of about 0.5 units on 067 each of the sugar rings.) The total van der Waals energy of about 048 -37kJ mol*t-1 *lbound acridine, is mainly due to 058 interactions between acridine and base-pairs, as expected. 057 When these calculations were repeated for protonated 069 9-aminoacridine, a higher interaction energy between drug and nucleic 036 acid was found than with proflavine, 059 and this may explain the greater anti-bacterial activity of 022 9-aminoacridine*S27*N. 062 However, we found that the interaction between 9-aminoacridine 064 and nucleic acid was greater with a Pritchard-type model for the 070 complex than with a Lerman-type model. To see why this was the case, 063 look once again at figure 5.4. Because the positive charge on 066 a greater electrostatic interaction, both with base-pairs and with 055 phosphate, is obtained in the Pritchard-type structure. 072 It is interesting to note that kinetic studies have suggested*S24 *Nthat 064 the binding of 9-aminoacridine to DNA takes place by a different 039 mechanism to the binding of proflavine. 057 The relatively high electrostatic interaction energy 056 found between proflavine and the nucleic acid base-pairs 056 warrants further discussion, because this is a source of 055 stabilisation whose importance appears not to have been 065 generally appreciated. Seeman's*S25 *Ncrystal structure for the 060 complex formed between 9-aminoacridine and ApU, in which the 059 nucleic acid had not associated into a double-helical form, 058 showed that an electrostatic interaction between phosphate 060 and acridine was not necessary for the formation of a stable 071 complex. The figures we gave in table 5.2 indicate that electrostatic 063 interactions with base-pairs are of the same order of magnitude 061 as the interactions with phosphate in a normal double-helical 056 nucleic acid, and correspondingly, this may provide some 053 explanation of how Seeman's structure was stabilised. 065 We investigated next the interaction between 9-aminoacridine 034 and single-stranded nucleic acids. 061 Pritchard, Blake and Peacocke*S16 *Nsuggested their model for 065 acridine intercalation partly because of the observed interaction 037 with single-stranded polynucleotides, 038 and we were interested to know whether 061 their model did give an appreciably more stable structure for 034 the complex than the alternatives. 070 Approximate structures for (9-aminoacridine.H*t+*l:dApA) were set 072 varying between 3*D? *Nand 35*D? *Nand with the acridine positioned in a 071 Lerman-type or Pritchard-type manner. We then refined the position of 065 the acridine cation, once again using the quasi-Newton algorithm. 021 The resulting optimum 067 interaction energies found are plotted in figure 5.7. We see that 061 complexing in the manner of Pritchard's model gives an energy 050 of interaction between dinucleotide and drug which 030 becomes more favourable as the 062 helix is wound up, and, bearing in mind the change in internal 077 energy of the dinucleotide with winding shown in *S?*N3.2.2, is probably most 066 favourable when the twist-angle between bases at the intercalation 066 site is approximately the same as in the uncomplexed nucleic acid. 057 The optimum structure found is illustrated in figure 5.8. 066 With the acridine complexed in a Lerman-type orientation, the 052 interaction energy is seen always to be less than in 072 the alternative Pritchard-type orientation. It is interesting to note, 065 though, that only about one third of the interaction energy found 060 for 9-aminoacridine with a double-stranded dinucleotide in a 065 Pritchard-type structure was lost on removal of one strand of the 067 helix; this is due to the asymmetry of the complex. In fact, the 062 optimum interaction energy found between 9-aminoacridine and a 071 single-stranded dinucleotide is somewhat greater than the corresponding 070 optimum for proflavine intercalated in a double-stranded dinucleotide. 067 Although we have calculated only one term in the total energy cycle 059 for intercalation, this result is certainly in keeping with 070 the similar binding constants found between single- or double-stranded 034 nucleic acids and acridines*S26*N. 074 The greater interaction energies calculated between certain acridines 067 and dinucleotides containing G-C base-pairs than with dinucleotides 066 containing A-T base-pairs was due entirely to the greater polarity 067 of G-C pairs than A-T. If an acridine was substituted in a manner 063 which produced a large local charge separation, then this could 064 interact favourably with a complementary distribution on the G-C 062 pairs. This is one possible reason for base-pair specificity 079 by intercalating agents; other reasons are discussed in *S?*N5.4 and *S?*N6.2. 002 *? *[Chapter 5.5*] 006 .tg NI 005 .ul 1 034 5.4 : Activity of Nitro-acridines. 006 .tg NA 034 The antibacterial activity of 061 the n-nitro-9-aminoacridines (n=1,2,3,4) is found*S27 *Nto be 038 unusually high, considering that these 043 compounds are only about 50*A> *Nprotonated 070 under the experimental conditions used to measure activity. Previous 054 calculations*S41 *Nshowed that the compounds possessed 074 relatively large charge separations on atoms around the nitro-group, which 076 could overlap favourably with complementary distributions on the base-pairs. 071 This would produce a high interaction energy with nucleic acid for both 064 protonated and unprotonated forms (see table 5.3 for some of the 063 interaction energies calculated) and we ascribed the biological 044 activity to these high interaction energies. 070 Further investigation of these compounds, however, has shown that this 052 may form only a part of the complete explanation for 069 the compounds' behaviour. In particular, note the following points: 006 .sp 1c 006 .ll -3 006 .in 10 006 .ti -5 005 .hy 3 077 1.ZZZThat, although each of the n-nitro-9-aminoacridines has an antibacterial 044 activity greater than that expected from the 084 compounds' pK*dA*l, this activity varied*S28 *Nas follows with 'n': 2=4*S<*N1*S<*N3. 006 .ti -5 077 2.ZZZThat recent studies*S29 *Non the anti-tumour properties of the compounds 073 n-nitro-9-(3'-dimethyl?amino?propyl?amino-)-acridine showed that activity 035 varied as follows: 2=4*S<*N3*Mj*N1. 006 .ti -5 053 3.ZZZThat a crystallographic study*S30 *Nperformed on 052 1-nitro-9-(3'-dimethyl?amino?propyl?amino-)-acridine 052 ("Ledakrin") showed that the compound was non-planar 072 (presumably due to the non-bonded contacts which would occur in a planar 064 compound between the nitro-group and the 9-amino group) with the 072 nitro-group twisted some 65*D? *Nout of plane and with the aromatic ring 049 system bent through 20*D? *Nabout C*T9*N-N*T10*N. 006 .ti -5 059 4.ZZZThat Ledakrin binds covalently to DNA *Iin vivo*S29*N. 006 .ti -5 081 5.ZZZThat the ease of polarographic reduction*S30 *Nof Ledakrin and its analogues 038 lies in the order 1*S>*N3*S>*N4*S>*N2. 006 .ti -5 059 6.ZZZThat reduction of aromatic nitro-compounds takes place 064 under anaerobic conditions in microsomes, by nitro-reductase and 059 utilising NADH as an electron donor, through hydroxylamino- 014 intermediates. 006 .ti -5 076 7.ZZZThat hydroxylamines are known to bind covalently to nucleic acid*S32*N. 006 .ll +3 003 .hc 005 .hy 1 005 .in 0 075 If the hydroxylamino derivatives of nitroacridines are responsible for 056 the anti-tumour properties of drugs related to Ledakrin, 050 then the approximate correlation found between the 082 Na*(*S?*Ni*)vely, the reason why, for example, the 3-nitro compound should be more 069 prone to reduction than the 2-nitro compound is illustrated in figure 066 5.10; an additional canonical form can be written for the 2-nitro 046 compound and this would be expected to lead to 058 greater stability of this compound, which would be lost on 039 addition of an electron (on reduction). 071 The non-planarity of Ledakrin in its crystalline state causes some 070 worry about the part intercalation can play in determining the effects 076 of this compound. If much of the bent aromatic system were to intercalate, 069 then it might be expected that a greater than normal extension of the 074 DNA helix would be required to accommodate it, and we have seen in chapter 073 3 that the helix does not possess sufficient flexibility for this. That 069 the nitro-group is also out-of-plane worries us less; because of the 005 .hc ? 071 adjacent dimethyl?amino?propyl?amino substituent, this group will have, 003 .hc 062 in any case, to be located in one of the grooves of the helix. 070 We performed molecular orbital calculations on the nitro-9-amino- 066 acridines using the CNDO/2 approximation, the results of which are 073 given in table 5.4. The calculated difference in energy between highest 074 occupied and lowest unoccupied molecular orbitals varied with the position 090 of the nitro-group as follows: 3 *S< *N1(ring bent) *S< *N4 *S< *N1(ring planar) *S< *N2. 076 This difference in energy has been used by other workers as a measure of the 073 ease of reduction of a compound (reduction being, to some extent, the end 052 result of adding an electron) and, for the compounds 066 substituted at positions 2,3 and 4, we see that there is a correct 071 prediction for the 1-nitro compound is incorrect, although it is better 046 with the ring system bent than with it planar. 073 The total energies calculated for the various molecules are similar, 070 with the exception of an entirely coplanar conformation of the 1-nitro 075 compound, which is shown to be some 1500 kJ mol*t-1 *ldestabilised relative 037 to the crystallographic conformation. 021 This 1-nitro compound 056 is found to be slightly more stable with the nitro-group 066 twisted out of plane, but with the ring system planar, than in the 071 crystallographic conformation, which might not perhaps be expected, but 077 the energy difference is small (being some 0.01*A> *Nof the total) and cannot 074 be expected to be accurate. The result does suggest, though, that little 051 energy penalty is paid for bending the ring system. 059 An intercalative binding of Ledakrin and its analogues 072 is therefore plausible, and the ease of reduction of the compounds could 034 explain their relative activities. 056 It only remains to discover why the anti-tumour activity 068 of Ledakrin is considerably greater than its 3-nitro analogue, since 070 experimental results show a relatively small difference in the ease of 072 reduction of the two compounds (and theoretical calculations predict the 075 opposite order*A?*N). To obtain an understanding of this, it is necessary 061 to consider what happens on covalent binding of hydroxylamino 017 compounds to DNA. 075 Relatively little is known about this reaction in the case of Ledakrin 047 and its analogues. However, related reactions 063 have been investigated (see Weisburger*S32 *Nfor a review); in 054 particular, Kriek*S33 *Nhas studied the interaction of 005 .hc ? 063 with DNA, and has found that attack occurs on the 8-carbon atom 037 of guanine, resulting in formation of 048 N-(deoxy?guanosine-8-yl)-2-acetyl?amino?fluorene 003 .hc 042 residues within the DNA (see figure 5.11). 077 We investigated the likely structures of intercalation complexes between 071 Ledakrin and its analogues and DNA, using the interactive graphics tool 071 described in chapter 4. It became clear that to obtain an appreciable 067 interaction between the molecules, it was necessary for the drug to 074 be placed in a Lerman-type complex; the long aliphatic sidechain attached 072 to the central ring of the acridine would prevent insertion of more than 069 one ring in a Pritchard-type complex. However, if the acridine were 066 inserted from the wide groove of the helix at an appropriate site, 055 a hydroxylamine group attached to C*T1 *Ncould approach 065 close to the guanine 8-carbon. On the other hand, a substituent 068 at C*T3 *Nof the acridine would be situated relatively far away from 069 this guanine atom, and it would be necessary to move the intercalated 073 acridine quite appreciably to improve the atoms' proximity. Figure 5.12 070 illustrates this point. (Intercalation from the narrow groove of the 071 helix would result in neither the 1- nor 3-carbon atoms in the acridine 066 being placed close to the 8-carbon in guanine, owing to the steric 034 effects of the deoxyribose groups. 047 The exceptional biological activity of the 071 n-nitro-9-aminoacridines and related compounds could be due to the high 073 interaction energy obtained when these compounds intercalate into nucleic 067 acid, both with protonated and unprotonated forms of the molecules. 043 It seems more satisfying, though, to assume 021 takes place, allowing 068 covalent binding to DNA to occur, and the relative activities of the 067 compounds would be determined partly by the ease of this reduction. 008 When the 071 structure of the intercalation complex is determined by the presence of 066 a bulky substituent on the aromatic nucleus, a hydroxylamino group 024 at some positions can be 046 considerably better disposed for reacting with 032 guanine residues than at others, 060 and, in the case of Ledakrin-like compounds, this results in 046 greatest activity with the 1-nitro derivative. 065 The ability to form a covalent linkage with one of the bases 063 in DNA gives us a second reason for base-pair specificity in an 020 intercalating agent. 002 *? *[Chapter 5.6.1*] 006 .tg NI 005 .ul 1 042 5.5 : Bisintercalation of Acridine Dimers. 006 .tg NA 080 In *S?*N1.1.1, we described the work performed by Le Pecq's group*S34 *Nand 076 Waring's group*S35 *Non the interaction of various acridine dimers with DNA. 072 These dimers contained two acridine residues linked through an aliphatic 069 chain connected to amino-groups on position 9 of the nuclei, although 069 different nuclei and different backbones were used by the two groups. 069 Le Pecq's group found that intercalation of both nuclei occurred only 068 with those compounds whose backbones were sufficiently long to allow 070 intercalation to take place at non-adjacent sites along the helix, but 068 Waring's group found that certain dimers bisintercalated even though 038 their backbone of six -CH*T2*N- groups 056 was only long enough to place the 9-amino nitrogen atoms 078 at most 8.8A apart. More recent results by Waring's group*S36 *Nsuggest that 060 compounds contained substituents on the acridine nuclei, for 045 the ability to bisintercalate was found to be 042 critically dependent on this substitution. 063 Acridine dimers with backbones containing five -CH*T2*N- groups 070 (which allows about a 7.5A separation of the aromatic nuclei) produced 065 strange results, showing properties consistent neither with their 069 being purely monofunctional nor with their being purely bifunctional. 063 Compounds with shorter backbones intercalated only one nucleus. 056 (See figure 1.7 for an illustration of these compounds.) 075 The neighbour-exclusion principle is violated by these acridine dimers 075 if the nuclei are stacked perpendicular to the helix axis in the complexes. 078 Our results in chapter 3 show strong theoretical evidence for the existence of 072 such an exclusion principle, at least in a continuing helix possessing a 074 fixed axis, and Bond's studies*S37 *Non a (terpyridine)-platinum(II) : DNA 068 complex showed experimentally that neighbour-exclusion was satisfied 013 in this case. 036 Also, if the acridine dimers violate 044 neighbour-exclusion, bisintercalation should 058 occur whenever their backbone allowed more than about 6.8A 059 separation of the aromatic nuclei and this is not observed. 059 Consequently, we should prefer some alternative explanation 042 for the behaviour of these compounds which 033 is consistent with other results. 044 We investigated a model for interaction 050 of these compounds with DNA which involved kinking 065 of the helix between two non-adjacent sites. This model allowed 051 intercalation of both nuclei of the acridine dimers 050 with two base-pairs sandwiched between the nuclei. 062 suggested that bending and kinking of the helix can take place 055 relatively easily and we felt that these conformational 049 changes would be limited more by the unfavourable 067 contacts produced between atoms in the base-pairs on either side of 071 the kink than by geometric limitations in the backbone. We have shown 068 in *S?*N3.2.2 that, with intercalation occurring at alternate sites, 064 the remaining links could extend so as to place their glycosidic 071 link atoms 5.4A apart. We assumed that this amount of extension could 030 still occur in a kinked helix. 052 Our model was relatively simple, but allowed us 059 to ask certain relevant questions concerning the energetics 052 of interaction of acridine dimers with nucleic acid. 059 A schematic representation of this model is given in figure 064 5.13. The precise position at which the kinking occurs is left 064 undefined; we merely assume that it takes place somewhere along 069 the deoxyribose-phosphate backbone linking the sandwiched base-pairs. 071 Clearly, the amount of kinking required to accommodate the intercalated 035 nuclei is limited by the non-bonded 049 contacts incurred between atoms in the base-pairs 074 *DA *Nand *DB*N. Also, it is limited by the degree to which the aromatic 076 nuclei can intercalate, since this insertion causes the aliphatic side-chain 065 of the acridine dimer to come into contact with the nucleic acid. 078 Smaller degrees of insertion require smaller angles of kink when other factors 067 remain constant, but are likely to be associated with a loss in the 056 stacking interactions between acridine and nucleic acid. 023 We wished to know: 006 .sp 1c 006 .ll -3 006 .in 10 006 .ti -5 020 degree of insertion. 006 .ti -5 073 2.ZZZWhat upper bound could be placed on an angle of kink by the approach 054 of atoms in the base-pairs on either side of the kink. 006 .ti -5 063 3.ZZZHow the kink required for bisintercalation varied with the 065 length of the acridine backbone and with the degree of insertion. 006 .ll +3 005 .in 0 047 The first point was studied by considering 050 the interaction of a dinucleotide fragment of DNA, 063 extended so as to allow intercalation in the normal way, with a 072 molecule of 9-ethylaminoacridine, protonated on the ring nitrogen. The 060 latter compound was placed in a conformation similar to that 058 which would be adopted by the acridine dimers of interest. 071 We then calculated the energy of interaction between these molecules as 039 the acridine was inserted from the wide 050 groove of the helix, using a gridsearch technique; 035 a contour diagram of this energy is 072 given in figure 5.14. The minimum energy is obtained when the distance 029 *Dd *Nof figure 5.13 is 6.9A. 068 (With 9-aminoacridine, inserting the ring system by another angstrom 064 resulted in a slight increase in stability, but the gain was not 075 significant.) Insertion from the narrow groove of the helix resulted in a 005 .ne 8 046 considerable loss in interaction energy*t***l. 005 .fn 1 066 *t** *lThe question of the direction of intercalation of acridines 070 into DNA is a vexatious one. For example*S39*N, the binding constant 060 between proflavine and DNA which is glucosylated in the wide 060 groove is similar to that with normal DNA, and this has been 062 interpreted as implying that attack is from the narrow groove. 067 the rate of intercalation (as opposed to binding) into glucosylated 039 DNA is much lower than into normal DNA. 005 .en 1 049 The relatively small difference found between the 059 interaction energies of protonated 9-aminoacridine and this 071 f_r_a_g_m_e_n_t_ of an acridine dimer with a dinucleotide is consistent 063 with Waring's results*S35 *Nwhich indicate considerably greater 063 binding constants between acridine dimers and nucleic acid than 068 for monomers, since the total interaction energy for the dimers will 027 be correspondingly greater. 002 *? *[Chapter 5.6.2*] 041 The maximum kink possible in the DNA 040 molecule was calculated using a model of 059 two adenine-thymine base-pairs stacked on one another. By 060 pivoting each base-pair about an axis joining the glycosidic 062 nitrogen link atoms (atom N*T9 *Nin the adenine group and atom 065 N*T1 *Nin the thymidine group), we calculated how the interaction 063 energy between the base-pairs varied with angle of pivot (which 063 is the same as the angle *Sq *Nin figure 5.13) as a function of 054 the separation of the glycosidic nitrogen atoms in the 069 base-pairs. The results are shown in figure 5.15 for the case where 061 the rotation axes in each base-pair were arranged parallel to 072 one another (that is, with 36*D? *Nunwinding in this part of the helix). 065 The important point to notice is the rapid rise in energy above a 069 given rotation angle for each separation of the base-pairs, caused by 035 atoms in the base-pairs approaching 064 one another more closely than the minimum in their van der Waals 056 potential curve. At 5.4A separation of the base-pairs, 043 15-17*D? *Nkink can occur on either side of 022 the base-pair sandwich 059 With smaller (and perhaps more reasonable) unwinding angles 048 at this point on the helix, the permissible kink 041 becomes smaller, until with no unwinding, 032 about 12-14*D? *Nkink can occur. 063 Finally, we considered what kink was required in the helix 062 to allow bisintercalation of an acridine dimer with a backbone 019 shorter than 10.2A. 035 The calculation was straightforward 060 (at least for the simple model considered here); elementary 027 trigonometry gives us that: 006 .sp 1c 005 .ce 1 043 *Dk *N= *Dq *N+ 2*Dd*Ntan*Sq *N- 6.72sec*Sq 008 *N.sp 1c 065 (where *Dk*N, *Dq*N, *Dd *Nand *Sq *Nare defined in figure 5.13). 052 Figure 5.16 shows how *Sq *Nchanges as a function of 068 *Dq *Nan*Dd *Nd with 5.4A extension of the non-intercalated residue; 053 maximum insertion of an acridine dimer whose backbone 040 is 8.8A long (with six -CH*T2*N- groups) 049 requires a kink of about 14*D? *Nwith this model. 059 These three calculations indicate that a model for the 060 bisintercalation of acridine dimers at non-adjacent sites is 068 reasonable. Whilst our representation of this model is simple, and 057 unlikely to be perfectly correct, it gives a satisfactory 064 explanation of the observed properties of these acridine dimers. 062 In particular, it explains why a dimer whose backbone contains 064 six -CH*T2*N- groups can bisintercalate without trouble, for the 055 kink required to obtain maximum insertion of the nuclei 058 can be obtained without causing undue strain in the helix. 036 The strange behaviour of the shorter 070 chain-length compounds*S35*N, containing five -CH*T2*N- groups, can be 061 explained by the fact that only very incomplete intercalation 060 of the two nuclei can occur without causing severe strain in 017 the nucleic acid. 063 We note that it is not necessary for the effects of such a 058 kink in the nucleic acid to be more than local. Sobell's 040 work*S40 *Non dinucleotide intercalation 065 complexes has indicated the possibility of another kink occurring 053 in the nucleic acid at the s_i_t_e_ of intercalation. 063 Although he too suggests that such kinking would be towards the 063 wide groove of the helix, we feel that small degrees of kink in 066 either direction are not ruled out, and it would be quite possible 069 for the various kinks to cancel one another out over several residues 013 of the helix. 062 An explanation can be given in terms of our model for the 060 lack of bifunctionality in the medium chain-length compounds 059 studied by Le Pecq's group*S34*N, which were substituted at 059 positions 2 and 6 on each nucleus with fairly bulky groups. 021 Since substituents at 068 these positions will approach the nucleic acid backbone more closely 066 than other atoms in the acridine nuclei, it is clear that a larger 059 total unwinding of the helix will be necessary at all three 055 residues involved in the interaction to accommodate the 054 acridine dimers than with the unsubstituted analogues. 048 Since the helix extension calculations described 059 in *S?*N3.2.2 showed that unwinding at an extending residue 063 was generally associated with a somewhat smaller winding at the 055 adjacent residue, we would expect that large amounts of 069 unwinding at consecutive residues would be unfavourable. This would 072 place steric constraints on the type of substituted acridine dimer which 021 could bisintercalate. 002 *? *[Chapter 5.7.1*] 003 .br 006 .tg NI 005 .ul 1 031 5.6 : Daunomycin Intercalation. 061 Crystal structure determinations on daunomycin analogues 070 have been performed on N-bromoacetyldaunomycin*S42 *N(using relatively 067 poor quality crystals), on carminomycin*S43 *N(4-hydroxydaunomycin) 065 and on a pyridine adduct of daunomycin*S44 *Nitself. These show 064 similar properties and differ only in fine detail, with the main 039 structural attributes being as follows: 006 .ll -3 006 .in 10 006 .sp 1c 006 .ti -5 058 1.ZZZThe cyclohexene ring is in a half-chair conformation. 006 .ti -5 106 2.ZZZThe torsional angles *Sc*N(C*T9*N-C*T10*N-O*T29*N-C*T30*N) and *Sy*N(C*T10*N-O*T29*N-C*T30*N-O*T31*N) 069 are about 246*D? *Nand 292*D? *Nrespectively in the crystal structure 067 of daunomycin, and are very similar in carminomycin. However, the 066 value *Sy *Nis about 20*D? *Ndifferent in N-bromoacetyldaunomycin, 068 indicating a slightly altered disposition of the daunosamine residue 028 to the rest of the molecule. 006 .ti -5 065 3.ZZZThe daunosamine residue is located in an approximately axial 069 conformation on the cyclohexene ring in daunomycin, whilst the acetyl 022 residue is equatorial. 003 .br 006 .ll +3 005 .in 0 062 The preferred conformation of the cyclohexene ring in solution 061 has been found, using nmr spectroscopy, to be the same as the 036 crystallographic conformation*S57*N. 066 One model for the intercalative binding of daunomycin to DNA, 053 proposed by Pigram and co-workers*S45*N, is based on 038 the crystallographic conformation, and 058 involves intercalation of the quinone ring system and also 061 electrostatic binding of the daunosamine amino-group to a DNA 068 phosphate residue separated by two base-pairs from the intercalation 067 patterns obtained from fibres of daunomycin:DNA complexes, the most 072 reasonable structure for the complex was obtained with 12*D? *Nunwinding 068 of the helix per intercalated residue, and the ability of this model 074 to explain the dominant nature of electrostatic forces on the interaction, 065 to predict the maximum observed binding of one molecule per three 069 base-pairs and to rationalise the effects of structural modifications 064 such as N-acylation of daunomycin on its binding, has lead to it 030 gaining widespread acceptance. 067 Another model, proposed by Henry*S46*N, for the interaction of 067 daunomycin with DNA involved the alternative "flipped" conformation 066 of the cyclohexene ring. By means of this conformational change, 067 the daunosamine residue is suitably placed to allow the amino-group 062 to interact electrostatically with the phosphate one base-pair 058 away from the intercalated ring system. Furthermore, the 049 hydroxyl-group on the cyclohexene ring is allowed 023 to form a hydrogen-bond 064 to the phosphate group adjacent to the intercalated chromophore, 060 whilst the hydroxyl-group on the daunosamine ring could form 061 a hydrogen-bond to phosphate two base-pairs away. A maximum 066 number of functionalities were therefore involved in this model of 063 the interaction. Although the conformation of the cyclohexene 060 ring required was not that found crystallographically, Henry 067 suggested that there would be little energy penalty associated with 057 the transformation, because of the comparable bulk of the 063 equatorial acetyl-group and axial sugar group, and that, in any 067 case, this would be recouped by the greater interaction possible in 037 his model between DNA and daunomycin. 066 than that found crystallographically is not necessarily important. 055 However, analogues of daunomycin possessing an inverted 020 configuration of the 063 daunosamine hydroxyl-group show binding and activity comparable 069 to daunomycin itself*S47*N, which suggests that the hydroxyl-group is 056 not involved the interaction, contrary to Henry's model. 063 We investigated firstly the effects of conformation on the 061 stability of the daunomycin molecule, and secondly the nature 061 of the interaction between daunomycin-like compounds and DNA. 006 .tg NI 005 .ul 1 062 5.6.1 : Variations possible in the Conformation of Daunomycin. 006 .tg NA 071 The effect of varying the two torsional angles *Sc *Nand *Sy *Nwas 065 studied by performing a grid-search type of calculation, in which 059 we calculated the energy of the daunomycin molecule at 8*D? 080 *Nincrements about each of the bonds C*T10*N-O*T29 *Nand O*T29*N-C*T30*N. This 053 energy was considered to arise from van der Waals and 063 electrostatic interactions between atoms in the sugar group and 057 those in the rest of the molecule, ignoring terms between 066 bonded and next-neighbour atoms. The way the energy changed with 059 conformation is shown in figure 5.17; contours are plotted 065 at 20kJ intervals and show the behaviour around the minimum only. 080 The torsional angles found in crystal structure determinations*S42*NZ*S43*NZ*S44 066 *Nare marked, and are seen to lie in the region of minimum energy. 065 Whilst the contours plotted in figure 5.16 show a relatively 065 flat area around the minimum, outside this area, the energy rises 058 extremely quickly, due to unfavourable non-bonded contacts 063 We feel confident, therefore, that the area of stability of the 063 daunomycin molecule, when the sugar group is placed in an axial 072 conformation, is limited to about a 60*D? *Nrange in the torsional angle 080 *Sc*N(C*T9*N-C*T10*N-O*T29*N-C*T30*N) and to about a 120*D? *Nrange in the angle 039 *Sy*N(C*T10*N-O*T29*N-C*T30*N-C*T31*N). 056 For interest, we performed a similar calculation on 064 the hypothetical compound 'Q' (illustrated in figure A1.1) which 066 is a 1,2,3,4-tetrahydrobenzacridine bearing a daunosamine residue. 065 We assumed that the compound would be protonated both on the ring 060 nitrogen and on the sugar amino-group. (If this is not the 069 case for the unsubstituted compound under biological conditions, then 069 there will undoubtedly be some substituted analogue for which it is.) 033 The resulting contour plot, shown 045 in figure 5.18, is compressed in terms of the 070 angle *Sy *Ncompared to daunomycin, owing to repulsion between the two 046 positive charges, one of which at least is not 063 particularly delocalised. It is also rather expanded in terms 067 of the angle *Sc*N, due to the lack of substituents on the acridine 059 nucleus. However, the interesting point is that the range 046 of conformations possible is not dissimilar to 041 the range possible for daunomycin itself. 002 *? *[Chapter 5.7.2*] 064 We then converted the standard daunomycin conformation into 037 a new form possessing an axial acetyl 057 residue (see figure 5.19), using the following algorithm: 006 .sp 1c 006 .ll -3 006 .in 10 006 .ti -5 094 1.ZZZConsider the atoms C*T9*N,C*T10*N,C*T11*N,C*T12*N,C*T13*N,C*T14 *Nin the cyclohexene ring 102 atom C8 at position *Dp_*T8*N, and also the points *Dq_*(*S1*T1*)0*N,*Dq_*(*S2*T1*)0*N, etc, which are 070 a unit distance along the side-chain bonds from atoms C*T10*N-C*T13*N. 006 .ti -5 052 2.ZZZDetermine the plane of the aromatic ring system 058 *Da_ *N= (*Dp_*T14*N-*Dp_*T9*N)*V^*N(*Dp_*T8*N-*Dp_*T9*N). 006 .ti -5 068 3.ZZZReflect *Dp_*T10 *Nto *Dp_*T13 *Nin the plane *Da_ *Nto produce 104 positions *Dp_*(*N'*T1*)0 *Nto *Dp_*(*N'*T1*)3 *Nand perform the same operation on *Dq_*(*S1*T1*)0 *Netc 052 to produce positions *Dq_*(*S1*T1*)*(*N'*T0*) *Netc. 006 .ti -5 131 4.ZZZThe new positions of atoms C*T10 *Nto C*T13 *Nare *Dp_*(*N'*T1*)0*N,*Dp_*(*N'*T1*)1*N,*Dp_*(*N'*T1*)2 *Nand *Dp_*(*N'*T1*)3*N. 006 .ti -5 079 5.ZZZThe group attached to atom C*I*dk *N*lat *Dp_*I*dk *N*lalong the direction 115 *Dq_*(*S1*I*dk*) *N*lhas rotated about an axis *Dq_*(*S1*I*dk*)*V*l^*Dq_*(*S2*I*dk*)*N*l', passing through a centre 160 *Dc_ *N= *Dp_*I*dk *N*l- *S*Dq_*(*S1*I*dk*)*N*l, where *Dp_*I*dk *N*l- *S*Dq_*(*S1*I*dk*) *N*l= *Dp_*(*N'*I*dk*) *N*l- *S*Dq_*(*S2*I*dk*)*N*l', to lie in the 096 direction *Dq_*(*S2*I*dk*)*N*l' from *Dp_*(*N'*I*dk*)*N*l. Perform these rotations to generate 065 the new positions of atoms in groups joined to the inverted ring. 005 .in 0 006 .ll +3 061 Evaluation of the energy difference between the standard 066 conformation of daunomycin and this "flipped" form of the molecule 065 was not easy, and it was necessary to make several approximations 063 before the calculation was possible. We adopted the following 010 procedure. 067 Firstly, since there was no guarantee that the most favourable 053 values for the torsional angles found in the inverted 064 structure would be those which had been produced by means of the 060 angles as variables and calculating the internal energy as a 060 sum of dispersion and electrostatic terms between non-bonded 066 and non-neighbour atoms. After twenty cycles of refinement using 064 the quasi-Newton algorithm, the internal energy had been reduced 064 by half and had become fairly stable, but was still considerably 060 higher than the corresponding value for the crystallographic 013 conformation. 059 We felt that the difference in internal energy between 062 the two conformations calculated using semi-empirical formulae 065 designed more for intermolecular interactions would be inaccurate 072 and consequently, we attempted to calculate this energy difference using 056 the CNDO/2 quantum approximation. Since the daunomycin 064 molecule contains more atoms than the available computer program 060 could handle, we studied a related molecule which lacked the 059 three aromatic rings of daunomycin, along with those groups 065 attached to them, and which contained merely the cyclohexene ring 065 and those groups attached to that ring. This molecule contained 068 forty four atoms, which was just within the capacity of the computer 008 program. 055 The total energies calculated for the molecule are 019 shown in table 5.5. 055 For both the protonated and unprotonated compounds, the 070 flipped conformation was found to be some 600 kJ mol*t-1 *lless stable 056 than its unflipped counterpart. However, the following 069 points must be borne in mind before this ridiculously large figure is 070 accepted as a true measure of the energy difference between the forms: 006 .sp 1c 006 .in 10 006 .ll -3 006 .ti -5 070 1.ZZZThe difference was only 0.1*A> *Nof the total energies calculated 058 that rounding error was limiting the precision attainable. 006 .ti -5 061 2.ZZZThere is no guarantee that the conformation generated by 056 the algorithm given above and subsequently refined using 060 semi-empirical potentials corresponded exactly with the true 069 minimum energy structure for the flipped form of the molecule; small 066 changes which might take place in the bond-angles and bond-lengths 058 could have relatively large effects on the internal energy 016 of the molecule. 006 .ti -5 064 3.ZZZThe CNDO/2 approximation*S53 *Nis known to exaggerate quite 064 grossly properties of molecules related to internal energy, such 028 as bond-stretching energies. 005 .in 0 006 .ll +3 063 Nonetheless, the result appears strange, even allowing for 070 possible gross errors in the calculation. After all, the sugar group 069 is at least as bulky as the acetyl group, and we might therefore have 059 expected that the opposite result would have been obtained. 054 The most rational explanation we can offer is that, in 077 the crystallographic structure, atoms O*T29 *Nand H*T23 *Nare well-positioned 062 for forming an intramolecular hydrogen-bond, which would close 086 the six-membered ring O*T29*N-C*T10*N-C*T11*N-C*T12*N-O*T23*N-H*T23*N, whereas, in the 042 "flipped" conformation, this cannot occur. 071 The existence of this hydrogen-bond has been suggested by Neidle*S44*N, 045 and, in carminomycin, the relevant proton has 046 been located in the required position for this 046 bond*S43*N. However, the energy of formation 073 of such a hydrogen-bond is unlikely to be more than about 25kJ mol*t-1*l, 063 over an order of magnitude less than the figure obtained above. 067 Other points, such as the fact that the amino-sugar group, adjacent 036 may have some bearing on the result. 060 It seems likely, on the basis of experimental evidence, 023 that daunomycin is more 058 stable with the cyclohexene ring in its crystallographical 070 conformation,. In this conformation, our results show that the sugar 061 group can adopt only a limited range of dispositions relative 066 to the aromatic ring system. Within the region of stability, the 070 internal energy appears well-behaved and relatively constant, which is 062 consistent with the different values of torsional angles found 052 in the crystal structure determinations performed on 050 daunomycin and its analogues. In particular, no 067 barriers are found separating one stable conformation from another, 066 which would hinder optimisation of the structure of daunomycin-DNA 022 complexes numerically. 064 The region of stability for a hypothetical daunoso-acridine 050 compound was similar to that found for daunomycin, 061 so that the existence of intercalation complexes between this 057 compound and DNA are not ruled out because of a different 023 preferred conformation. 002 *? *[Chapter 5.7.3*] 003 .br 006 .tg NI 005 .ul 1 047 5.6.2 : Interactions between Daunomycin and DNA 006 .tg NA 053 The results of *S?*N5.6.1 led us to believe that 048 Pigram's model*S45 *Nfor the interaction between 061 daunomycin and DNA was more likely to be correct than Henry's 078 model*S46*N, and we based further studies on the crystallographic conformation 014 of daunomycin. 060 The interaction between daunomycin and nucleic acid was 065 studied in two stages. Firstly, we examined the molecules using 059 obtained approximate structures for the complex, these were 049 refined numerically on the basis of their energy. 069 Neidle's*S44 *Ncoordinates were used for the daunomycin molecule 043 and we studied the interaction of this with 060 nucleic acid fragments comprising a dinucleotide extended on 063 one chain by the sequence pdAp-, as illustrated in figure 5.20. 060 This was the smallest fragment which could illustrate models 058 similar to Pigram's without omitting potentially important 061 interactions, and was chosen so as to economise on the amount 066 of computation necessary. The nucleic acid structures were built 064 up from extended trinucleotide conformations obtained in chapter 036 3, with the terminal phosphate group 039 possessing a normal B-DNA conformation, 073 and were unwound at the intercalation site by between 5*D? *Nand 33*D?*N. 059 It will be recalled that the process of helix extension was 067 accompanied by a change in the twist of the helix at more than just 070 the intercalation site, so that these models corresponded to fragments 046 of DNA unwound by between -3*D? *Nand 25*D?*N. 057 The problems encountered when setting-up approximate 069 daunomycin:DNA complex structures using interactive computer graphics 067 have been discussed in *S?*N4.5. We had had particular difficulty 069 in finding the best interlock between the sugar residue of daunomycin 065 and the deoxyribose-phosphate backbone of DNA, owing to the large 036 number of atoms in the complex, and, 038 we chose to base further refinement on 064 structures which contained no obviously bad interactions, but in 063 which the amino-group of daunomycin and the appropriate nucleic 047 acid phosphate group were relatively far apart. 069 5.21, were based on a Pritchard-type*S16 *Norientation of the quinone 040 ring system between the base-pairs, with 044 insertion from the wide groove of the helix. 064 Adopting a Pritchard-type orientation for the initial model 068 meant that the methoxy-group at position 4 on the daunomycin nucleus 066 was inserted through the nucleic acid and lay in the narrow groove 052 of the helix. Pigram's model, however, placed this 055 group in the wide groove of the helix, by orienting the 065 quinone ring system more towards a Lerman-type*S15 *Norientation. 058 The two structures are not interconvertible by a numerical 064 optimisation, because there will be an energy barrier to passing 061 the methoxy-group through the helix. We do not believe that 057 this barrier will be large, since the methyl-group is not 061 very bulky, and the helix extension calculations described in 063 *S?*N3.2 showed that slightly more than a 6.72A spacing between 058 base-pairs could be achieved without causing undue strain. 067 The greater potency observed*S54 *Nfor the 4-methoxy derivatives of 066 both daunomycin and adriamycin (being as effective as their parent 064 compounds, but at up to ten times lower dose) could be explained 060 in terms of a larger activation energy to binding the parent 065 compounds than to binding of these derivatives, and this would be 060 expected if binding requires penetration of the helix by the 026 intercalating chromophore. 062 Ideally, we should have liked to compare the properties of our 065 model with those of Pigram's, but unfortunately the computational 023 expense ruled this out. 070 Also, because of this computational expense, it was necessary to treat 070 the nucleic acid molecule as a rigid unit, whose conformation had been 063 not realistic, but we hope that the errors incurred in this way 066 were not large. The assumption reduced the amount of calculation 068 required by at least half, since it eliminated the need to calculate 056 interaction terms between atoms within the nucleic acid. 070 We refined the approximate structures in three stages. Firstly, 067 we treated the daunomycin as a rigid unit, which could only move or 071 rotate *Ien masse*N, and we performed sufficient cycles of optimisation 062 using the quasi-Newton algorithm*S2 *Nto place the molecule in 067 an energetically-favourable position. This occurred very quickly, 067 and some ten line-searches were sufficient to reduce the derivative 022 of the energy function 006 .tg NI 060 in terms of the degrees of freedom available to the molecule 006 .tg NA 068 to negligible proportions. Due to the combination of a potentially 067 very strong electrostatic interaction between sugar amino-group and 070 nucleic acid phosphate two base-pairs away from the intercalation site 062 and the fact that the formulation used to calculate dispersion 061 and overlap energies did not indicate any great difficulty in 071 bringing planar aromatic systems somewhat closer than the 3.36A spacing 062 found crystallographically, the entire daunomycin molecule was 061 pulled towards this phosphate, moving the quinone ring-system 050 until it was less than 3A away from one base-pair. 065 The next stage of refinement treated the daunomycin molecule 070 as a set of rigid groups connected through bonds about which rotations 069 could occur, and attempted to minimise the interaction energy between 067 drug and nucleic acid, in terms of the torsional angles about these 063 quasi-Newton algorithm. This time, the optimisation proceeded 065 erratically, sometimes locating the minimum along lines of search 067 very quickly and sometimes wasting a great many energy evaluations. 066 Investigation showed that this was due to an inconsistency between 062 the values calculated for the energy and its derivatives; the 064 minimum energy value along a line of search did not occur at the 072 point at which the gradient along that line was zero. This meant that, 064 if the optimisation algorithm selected a point which lay between 071 that with minimum energy and that with zero gradient, the interpolation 042 routine would fail, and movement along the 029 line of search would be slow. 002 *? *[Chapter 5.7.4*] 065 That the behaviour of the routine was sometimes good was due 061 to the fact that the routine did not attempt to locate minima 070 along lines of search precisely; if a point was found which satisfied 070 Wolfe's criterion*S6 *N(gradient along line of search reduced by half) 066 before the region of inconsistency was reached, then behaviour was 070 perfectly satisfactory. Since the discrepancy between energy minimum 056 and gradient zero was generally very small (of the order 072 of 0.01*D? *Nrotation about each bond), this termination condition could 024 frequently be satisfied. 066 Thorough testing showed that the discrepancy between function 063 and gradient occurred only when the degrees of freedom included 061 rotations. Whenever the variables involved merely cartesian 060 coordinates, the minima found from the two quantities agreed 058 to within machine accuracy. We were completely unable to 067 find out why this happened; the algebra derived seemed correct and 005 .ne 6 045 no errors could be found in the coding*t***l. 028 *t** *lIt seems particularly 058 galling that the difficulties arose because of attempts to 059 simplify the optimisation. Treating the molecule as a set 061 of units connected through rotatable bonds allows most of the 058 degrees of freedom available to the molecule, whilst still 059 retaining a managable number of variables. Because of the 066 potential utility of this method of structural refinement, we give 062 the algebra used in figure 5.22, in the hope that someone will 014 see the error. 005 .en 1 069 Because the discrepancy between energy minimum and gradient zero 063 was small, the refinements did in fact result in an appreciable 061 inprovement in the interaction between daunomycin and nucleic 054 acid, although they used some three to five times more 057 times more computer-time than should have been necessary. 064 Some form of order appeared in the relative interaction energies 061 with nucleic acids possessing different unwinding angles, and 067 the close approach of the quinone ring system to a base-pair, which 060 occurred in the first stage of refinement, was alleviated to 055 some extent by the adjustment in the relative positions 038 of the sugar group and aromatic system 065 of daunomycin which could occur through rotations about the bonds 036 C*T10*N-O*T29 *Nand O*T29*N-C*T30*N. 066 Finally, we allowed each atomic coordinate to vary within the 066 daunomycin molecule. This meant that there were some two hundred 059 variables, and the refinement was therefore performed using 032 a conjugate direction algorithm. 031 Some fifty linear searches were 048 performed on each complex structure, after which 069 the rounding error incurred in energy and gradient evaluation was too 063 At this stage, the energies obtained appeared to have converged 064 rather well, with each of the last few linear searches producing 052 less than a 1kJ mol*t-1 *limprovement in energy, and 054 individual terms in the gradient vector were typically 063 about 2 kJ mol*t-1 *lA*t-1*l, indicating that the complexes had 065 reached relatively flat areas in their potential energy surfaces. 060 In figure 5.23, we show the variation in the calculated 065 optimum interaction energies between daunomycin and nucleic acid, 067 as a function of the total helix unwinding. (The variation in the 057 internal energy of daunomycin in the different structures 061 has little effect on this graph.) The most stable structure 063 found was unwound by 13*D?*N, although the energy of the system 048 remained fairly constant in the range 8-16*D?*N, 024 with a 21*D? *Nunwinding 073 at this site being accompanied by an 8*D? *Nwinding at the adjacent site. 070 This figure of 13*D? *Nis almost identical with the 12*D? *Nwhich 066 is obtained from analysis of x-ray diffraction data*S44 *Nand from 045 measurements of supercoil unwinding*S20 21*N, 053 a point which we find particularly satisfying in view 055 of the computational expense involved in performing the 044 numerical refinements described above*t***l. 005 .fn 1 064 *t** *lEach point marked on figure 5.23 required about two hours 027 computer-time to determine. 005 .en 1 057 The optimum refined structure was characterised by having 068 the anthraquinone ring system some 4.3*D? *Nout of the plane defined 059 by the nucleic acid helix axis along the length of the ring 064 system, and about 12*D? *Nout of plane across its breadth. Had 068 optimisation, this would probably have been accommodated by twisting 063 the base-pairs in the vicinity of the interaction, a structural 061 change which has been found to involve little energy penalty. 058 Slight adjustment had taken place in some torsional angles 060 within daunomycin, mainly as a result of the second stage of 045 refinement, and we list these angles in table 004 5.6. 056 Table 5.7 shows how the total interaction energy of 066 some -460kJ mol*t-1 *lbetween daunomycin and nucleic acid was made 061 up. Whilst the true enthalpy of interaction between the two 071 systems will only be a small fraction of this internal energy*V*t*N*l, 005 .fn 2 077 *V*t *N*lQuadrifoglio*S48 *Nhas measured the enthalpy of interaction between 059 daunomycin and calf thymus DNA to be about -27kJ mol*t-1*l. 005 .en 2 060 owing to the other terms involved in a complete energy cycle 065 which we did not consider, the breakdown reveals some interesting 007 points. 002 *? *[Chapter 5.7.5*] 060 Firstly, the dominance of the electrostatic interaction 058 between phosphate and amino-sugar is evident, in agreement 064 with previous suggesions*S45*N. This is almost entirely due to 052 the close approach of positively-charged nitrogen to 064 negatively-charged phosphate oxygen. The distance N*S..*NO was 057 3.4A, which is somewhat longer than would be expected for 059 a hydrogen-bond distance, but rotation of the the phosphate 057 group, a conformational change which we did not consider, 039 would undoubtedly reduce this distance. 061 The interaction between the anthraquinone ring system of 053 daunomycin and nucleic acid was made up of a moderate 053 term. In view of the lack of a formal charge on the 053 anthraquinone, the electrostatic contribution must be 063 due to a subtle interplay of the residual atomic charges in the 060 two systems. This contribution was, however, less than the 066 corresponding term found with an intercalating protonated acridine 059 (see table 5.2). We had ascribed much of the stability of 060 acridine intercalation complexes to this electrostatic term, 065 in particular to the interaction between acridine and base-pairs, 060 and it is clear that daunomycin intercalation (as opposed to 066 external binding to DNA) cannot be stabilised by the same factors. 061 However, the dispersion energy term between anthraquinone and 061 nucleic acid was much greater than the corresponding term for 074 acridine intercalation (-79kJ mol*t-1 *las against about -35kJ mol*t-1*l), 057 which means that the total interaction between the planar 060 intercalating section of the daunomycin molecule and nucleic 058 acid is very comparable to the total interaction found for 068 the acridines (-127kJ mol*t-1 *las against -90 to -150kJ mol*t-1*l). 057 The extra dispersion energy was caused by the larger size 059 of the anthraquinone ring system (four rings as against the 064 acridines' three) and by the presence of heavy atom substituents 020 on this ring system. 055 We see here a possible explanation for the lack of 064 anti-tumour activity in daunomycin analogues lacking a four-ring 060 system*S46*N. Unable to stabilise an intercalation complex 048 by electrostatic interactions in the same way as 063 the acridines or ethidium, daunomycin relies instead on a large 056 dispersion energy contribution, and this will be reduced 059 if the size of the intercalating moiety is reduced. It is 062 diffraction patterns for daunomycin:DNA complexes changed with 052 humidity, which was taken to indicate a considerable 066 hydrophobic contribution to the interaction, and this is certainly 058 in keeping with the large dispersion energy which we find. 061 Various views of the optimum refined structure are given 062 in figure 5.24. It can be seen that only a group attached to 064 C*T34 *Nin the manner of the daunomycin amino-group can approach 056 the appropriate phosphate group on DNA at all closely in 058 an intercalation complex. No other group of atoms in the 058 system appears to be quite so important in determining the 058 interaction, with the possible exception of the daunomycin 060 O*T23 *Natom, which appears to stabilise the conformation of 021 the cyclohexene ring, 041 allowing the amino-phosphate interaction. 056 We investigated the interaction of three daunomycin 063 analogues with the nucleic acid fragment previously considered. 044 Firstly, replacing the methoxy-group at C*T4 032 *Nby hydrogen resulted in a loss 052 of interaction energy of some 10kJ mol*t-1*l, mainly 068 due to the disappearance of the dispersion energy terms between this 065 methoxy-group and nucleic acid. The demethoxy-daunomycin showed 072 no tendency on refinement to withdraw itself from the intercalation site 060 and rotate round from its almost-Pritchard*S16 *Norientation 034 towards a Lerman*S15 *Norientation 060 and we must therefore conclude that the model set up for the 059 interaction with daunomycin was not prevented from changing 067 to a more favourable Lerman-type model because of an energy barrier 028 caused by the methoxy-group. 058 The next analogue studied was an epimer of daunomycin 060 This placed the amino-group further away from phosphate, and 063 there was a consequent loss of about 25kJ mol*t-1 *linteraction 058 energy, as expected. Refinement of the structure did not 036 lead to any significant improvement. 070 Thirdly, we investigated the interaction between nucleic acid and 067 the hypothetical daunoso-acridine compound discussed in *S?*N5.6.1. 037 Since this compound was assumed to be 054 protonated both on the amino-sugar and on the aromatic 059 ring system, and since its preferred conformation was found 060 to be similar to that of daunomycin, it was expected that it 041 would complex strongly with nucleic acid. 047 The energy of interaction between the molecules 074 was found to be about 25*A> *Nthan the corresponding value for daunomycin, 035 mainly due to the new electrostatic 062 contributions between the aromatic ring system of the acridine 060 and the phosphate groups and base-pairs of the nucleic acid. 069 Finally, we calculated the interaction energy between daunomycin 064 and nucleic acid fragments with different base sequences to that 059 used previously. It was not possible to perform more than 061 a cursory survey because of the computational expense, and we 058 were limited to investigating three analogues. Using the 054 optimum structure found for previously for the nucleic 067 acid : daunomycin complex (with 12*D? *Nunwinding of the helix), we 025 changed some of the bases 050 and then refined the daunomycin coordinates again. 054 Little movement occurred, and the interactions between 061 daunomycin and those parts of the nucleic acid fragment which 062 were unchanged were almost identical to their previous values. 065 present in the nucleic acid, because of the greater electrostatic 060 interaction with the more polar base, but decreased if there 021 was a pyrimidine base 053 between the amino-phosphate bond and the intercalated 066 chromophore, on the same side of the helix as the amino-phosphate. 061 In particular, a thymine group caused a large destabilisation 025 in this position, whereas 049 cytosine in this region caused a somewhat smaller 050 destabilisation. This was due to steric effects. 061 The interaction calculated between daunomycin and each of the 060 bases in the nucleic acid fragments is given in figure 5.25. 073 Care must be taken in interpreting these results, for it is possible 064 that refinement merely moved the daunomycin into a local minimum 070 of high energy, rather than finding a new configuration of low energy. 073 The results suggest, for example, that binding would occur preferentially 073 into G-C rich DNA, which has been observed in some studies*S49*N, but not 060 in others*S50*N. It also suggests, though, that binding to 068 poly(dA).poly(dT) would be stronger than to poly(dA-dT).poly(dA-dT), 051 because steric hindrance caused by pyrimidine bases 034 would always occur with the latter 060 polynucleotide, regardless of the position of intercalation, 061 and in fact the opposite result is observed*S51*N. However, 071 poly(dA).poly(dT) is known to adopt a non-B-DNA conformation*S52*N, and 063 it is obvious that results obtained from nucleic acid fragments 065 derived from the B-DNA structure are inapplicable to this system. 002 *? *[Chapter 5.8*] 006 .tg NI 005 .ul 1 014 5.7 : Summary. 006 .tg NA 066 It is clear that the use of automatic optimisation algorithms 063 their internal energy using semi-empirical potential functions, 065 can allow reasonable predictions to be made of some properties of 060 the complexes. For example, the predicted unwinding angles 058 which occur in nucleic acid on intercalation of proflavine 060 or daunomycin agree well with the experimentally-determined 049 values. The method also allows an understanding 051 of why some properties are observed experimentally. 060 9-aminoacridine and proflavine appear to bind to DNA through 061 different mechanisms, and this could be explained in terms of 068 the preference shown by the former for Pritchard-type configurations 061 and by the latter for Lerman-type configurations. The total 049 stacking energy between intercalating chromophore 064 and base-pairs is apparently as important in determining whether 066 or not the interaction will occur as the electrostatic interaction 060 with phosphate, and it seems that either an electrostatic or 063 a dispersion contribution to this stacking may dominate. This 064 is demonstrated by the breakdown of total interaction energy for 073 proflavine and daunomycin, and partly confirmed by the non-double-helical 064 structure found by Seeman for the complex (9-aminoacridine:ApU), 062 and by the lack of biological activity in daunomycin analogues 034 lacking the four-ring chromophore. 058 The delocalisation of charge caused by substituents on the 068 intercalating ring systems is probably more important in determining 062 strength of interaction than is the potential ability of these 060 substituents to hydrogen-bond to DNA; thus the amino-groups 064 in proflavine (and presumably also in ethidium, although we have 058 not studied this) induce delocalisation of positive charge 049 present on both nucleic acid bases and phosphate. 062 However, it is also clear that the method is fraught with 055 difficulties. The computation required to perform the 043 optimisations was so large that only a very 059 small proportion of the calculations required to survey the 059 properties of intercalation complexes could be carried out. 026 We based most calculations 066 on one nucleic acid fragment, whilst there are ten distinguishable 062 dinucleotide fragments of DNA alone. Equally, we showed that 056 a modification to Pigram's model for daunomycin binding, 058 with a Pritchard-type insertion of the chromophore instead 063 of a Lerman-type insertion, provided a satisfactory explanation 057 of some of the observed properties of the drug. We were 055 not, however, able to show that Pigram's original model 059 did not provide an equally good explanation. Nor could we 073 demonstrate the stronger binding of daunomycin to poly(dA-dT).poly(dA-dT) 065 than to poly(dA).poly(dT), because our calculations did not allow 068 for variations in nucleic acid conformation caused by these sequence 012 differences. 063 It seems that most of the limitations apparent in the work 059 arose because of the computational expense of the numerical 059 refinements. Shortcomings in the methods used to estimate 060 conformational stability were not normally critical, because 069 most of the systems possessed well-defined minimum energy structures. 067 Energy calculations become less reliable when there are alternative 067 structures or conformations of similar energy, because the accuracy 065 of the potential functions, or quantum mechanical approximations, 056 may well be insufficient to predict the correct minimum. 002 *? *[Chapter 5 Refs and Tables*] 007 .sp 10c 003 .nj 006 .tg NA 006 .ls 2c 005 .ul 1 033 1. Conjugate Direction Algorithm. 068 Linesearch Function Calls Energy after *MU*NGradient*MU 065 *NMade Search after Search 061 0 1 -117.3 30.92 062 1 3 -126.8 7.946 061 2 5 -130.4 19.89 062 3 7 -132.8 4.025 061 4 9 -137.4 14.31 062 5 12 -138.5 3.691 062 6 14 -138.8 3.176 062 7 16 -139.1 2.308 062 8 18 -139.1 0.857 062 9 21 -139.1 0.393 062 10 24 -139.1 0.130 062 11 27 -139.1 0.084 062 12 30 -139.1 0.018 062 13 33 -139.1 0.006 062 14 36 -139.1 0.002 005 .ul 1 026 2. Quasi-Newton Algorithm. 068 Linesearch Function Calls Energy after *MU*NGradient*MU 065 *NMade Search after Search 061 0 1 -117.3 30.92 062 1 3 -126.8 7.946 061 2 5 -130.4 16.20 061 3 7 -131.7 24.44 062 5 10 -139.1 1.371 062 6 11 -139.1 0.770 062 7 12 -139.1 0.404 062 8 13 -139.1 0.206 062 9 14 -139.1 0.073 062 10 15 -139.1 0.013 062 11 16 -139.1 0.001 006 .tg NI 005 .ul 1 049 Table 5.1 : Behaviour of Optimisation Algorithms. 006 .ls 3c 003 .bp 006 .tg NA 062 Dinucleotide section E(vdw) E(elect) kJ mol*t-1 008 *l.sp 1c 047 Deoxyribose group 1X -2.4 55.3 047 2X 4.9 84.0 047 1Y -2.4 74.7 047 2Y -1.4 44.0 006 .sp 1c 047 Phosphate group X -0.9 -106.4 047 Y -0.2 -76.7 006 .sp 1c 047 Bases : Adenine 1X -9.8 -53.2 047 2X -4.7 -28.9 047 Thymine 1Y -12.4 -27.7 047 2Y -8.1 -16.0 006 .tg NI 005 .ul 2 009 Table 5.2 067 Contributions to Energy of (Proflavine.H*A*t+ *N*l: d(A-A).d(T-T)). 003 .bp 006 .tg NA 031 1. Molecular orbital energies. 060 HOMO LUMO *SD 067 *N1-nitro 9-aminoacridine (1) -0.3547 0.0135 0.3682 au 065 (2) -0.3567 0.0161 0.3728 au 065 2-nitro 9-aminoacridine -0.3637 0.0126 0.3763 au 065 4-nitro 9-aminoacridine -0.3615 0.0084 0.3699 au 018 2. Total energies 046 1-nitro 9-aminoacridine (1) -449272 kJ 046 (2) -449312 kJ 046 (3) -447832 kJ 046 2-nitro 9-aminoacridine -449302 kJ 046 3-nitro 9-aminoacridine -449293 kJ 046 4-nitro 9-aminoacridine -449301 kJ 067 1) Crystallographic conformation for Ledakrin, viz -NO2 group 65*D? 048 *Nout of plane and 20*D? *Nbend about C9-N10. 060 2) -NO2 group 65*D? *Nout of plane; aromatic system planar. 009 3) Planar 006 .tg NI 005 .ul 2 009 Table 5.4 045 Various Energies of n-nitro 9-aminoacridines. 003 .bp 006 .tg NA 067 Compound Optimum energy (kJ mol*t-1*l) 064 Unprotonated Protonated 061 Acridine -142 -344 061 9-aminoacridine -145 -331 061 1-nitro 9-aminoacridine -193 -383 061 2-nitro 9-aminoacridine -201 -377 061 3-nitro 9-aminoacridine -178 -357 061 4-nitro 9-aminoacridine -206 -346 006 .tg NI 005 .ul 3 009 Table 5.3 060 Comparison of Optimum Interaction Energies found in Previous 047 Work*S41 *Nbetween Acridines and Dinucleotides. 062 (Calculations based on sum of electrostatic interactions, 066 calculated using PPP-determined*S55 *Natomic charges for the 059 acridines, and van der Waals interactions, calculated 045 using Giglio's*S56 *Nparameterisation.) 003 .bp 006 .tg NA 058 1.ZZZUnflipped, not protonated on amino-group -582203 kJ 058 2.ZZZFlipped, not protonated -581557 kJ 059 ZZZZZ(Difference between 1 and 2 646 kJ) 058 3.ZZZUnflipped, protonated on amino-group -580826 kJ 058 4.ZZZFlipped, protonated -580160 kJ 059 ZZZZZ(Difference between 3 and 4 666 kJ) 006 .tg NI 005 .ul 2 009 Table 5.5 037 Energies of Daunomycin Conformations. 003 .bp 006 .tg NA 051 Before After 065 Angle 1 2 3 4 5 6 7 109 C*T9*N-C*T10*N-O*T29*N-C*T30 *N246.2*D? *N240.0*D? *N240.5*D? *N237.5*D? *N241.6*D? *N227.8*D? *N232.0*D? 111 *NC*T10*N-O*T29*N-C*T30*N-O*T31 *N292.5*D? *N271.8*D? *N290.1*D? *N294.9*D? *N291.4*D? *N308.5*D? *N302.4*D? 008 *N.tg NI 005 .ul 3 009 Table 5.6 059 Torsional Angles in Daunomycin, before and after Refinement 033 of Drug : Nucleic Acid Complexes. 047 (1) Crystal Structure of Daunomycin*S44*N. 054 (2) Crystal Structure of N-bromodaunomycin*S42*N. 049 (3) Crystal Structure of Carminomycin*S43*N. 046 (4) With nucleic acid unwound by 17*D?*N. 046 (5) With nucleic acid unwound by 13*D?*N. 045 (6) With nucleic acid unwound by 9*D?*N. 045 (7) With nucleic acid unwound by 5*D?*N. 003 .bp 006 .tg NA 006 .ls 2c 057 Anthraquinone Daunosamine 029 Dispersion + Overlap Energies 054 Sugar groups residue 1 -7.99 -0.08 054 2 -9.97 -0.23 054 3 -1.38 -1.78 054 2 -29.24 -3.22 054 3 -1.12 -0.24 054 Phosphate residue 1 -1.19 -0.24 054 2 -0.52 -0.09 054 3 -0.20 -1.15 064 Total -79.06 -7.40 -86.46 022 Electrostatic Energies 054 Sugar groups residue 1 14.80 46.33 054 2 -12.69 58.65 054 3 6.57 78.50 054 Bases residue 1 -22.60 -12.38 054 2 -3.85 -19.60 054 3 -2.41 -44.13 054 Phosphate residue 1 -9.82 -82.85 054 2 -9.60 -68.10 054 3 -8.41 -276.59 064 Total -48.01 -320.17 -368.18 006 .ls 3c 006 .tg NI 005 .ul 3 009 Table 5.7 058 Contributions to Interaction Energy between Daunomycin and 042 Nucleic Acid (13*D? *NUnwinding of Helix). 034 (All figures in kJ mol*t-1*l) 003 .bp 006 .ls 3c 006 .tg NI 005 .ul 1 025 References for Chapter 5. 006 .tg NA 060 1. 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