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The possible role of solitonic processes during A to B conformational changes in DNA
THE POSSIBLE ROLE OF SOLITONIC PROCESSES DURING A TO B CONFORMATIONAL CHANGES IN DNA n
ANGSHUTOSH KHAN Ashutosh College, Calcutta, India.
H DEBAJYOTI BHAUMIK Bose Institute, 93/l Acharya PrafuIIa Chandra Road, Calcutta-700009, India. n
BINAYAK DUTTA-ROY * Saha Institute of Nuclear Physics, 92 Acharya Prafulla Chandra Road, Calcutta-700009, India.
the conformations which the DNA molecule can adopt, the transition beween the A and B families, controlled by water content (relative humidity), seems to be implicated in the transcription process. Focusing on the main structural difference involved (tilting of base normals with respect to the helix axis), a model is constructed, solitary wave solutions of the resulting equation of motion are demonstrated and possible experimental implications indicated. Among
1. Zntroductiorl. After the double-helical structure of DNA was revealed in 1953 by Crick and Watson, subsequent studies of X-ray diffraction patterns from drawn fibres of DNA under diverse salt, alcohol and humidity conditions together with investigations on circular dichroism, infra-red spectrum and use of other solution techniques, led to the realization that the double-helical DNA structures can be categorised into two right-handed families-A and B-with fundamental differences as to the manner in which the base-pairs are stacked and the sugar-phosphate back-bone strands are wrapped along the helix axis. Furthermore. an unexpected left-handed Z-helix was found from X-ray analysis of single crystals. made possible by the development of tri-ester methods for the synthesis of pure DNA with a predetermined sequence. An excellent review of the anatomy of A-, B- and Z-DNA can be found in an article by Dickerson et al. (1982). wherein the salient structural distinctions are discussed. For some of the existing ideas *: To whom correspondence
should be addressed.
783
784
A. KHAN
etal.
and speculations regarding the functional roles of the different families of DNA conformations one may refer to the inexpensive textbook by Volkenstein (1983). Confining our attention, for the present purpose, to the A and B families, it may be discerned that the principal difference here resides in the inclination angle (4) of the base normals to the helix axis, being -20” and -0’ for the two forms, respectively, with a concommitant change in conformation of the sugar from a 3-endo to 2-endo, and a shift in the pitch. At low relative humidity (up to 70%) DNA crystallises in the monoclinic A form while at high humidity the hexagonal B form is obtained. The normal (or resting) state of DNA is presumably the B form and the mechanism and kinetics of its transformation to the A form is of considerable functional importance since this metamorphosis is very likely implicated in the process of transcription, whereby information stored in DNA through the sequence of bases, is conveyed to RNA (through the synthesis of RNA on the DNA template with the aid of RNAgpolymerase), which in turn transmits the genetic information to the ribosomes on which biosynthesis of proteins is accomplished. The propagation of localised excitations (or conformational changes) along macromolecules (or their aggregates) is often considered to play an important role in transport and transduction processes and in the kinetics of conformational transformations occurring in biology. The antagonists of this viewpoint, however, draw attention to the smallness of the lifetime of such excitations, which would indeed be the case if these were associated with local chemical groups, on the other hand stands the contention that the existence of collective modes with concommitant non-linearities in such systems makes possible relatively stable localised excitations through solitary wave solutions. Indeed Davydov (1982) has emphasised the possible role of solitons in a-helical protein molecules in their function, such as, for instance, their implication in the contractile activity of the actin-myosin complex in muscle tibres. The present authors too have sought (Bhaumik et al., 1982) for a solitonic excitation in macromolecular systems within the context of a model proposed by Frohlich (1973) involving polar modes and attendant elastic restoring forces. Englander et al. (1980) have predicted the possibility of a solitonic mode in the helix-coil transition involved in melting of DNA. Jensen et al. (1983) have invoked soliton-like processes during salt-induced right-left conformational changes in DNA, while Balanovski and Beaconsfield (1982) have considered, in like manner, the question of the A + B transformation. In the latter paper, however, attempts have not been made to categorise the structural features involved in the process. The object of the present study is to focus attention on the main structural distinction between A- and B-DNA and thereby to attempt a more concrete realisation of solitonic processes in the kinetics of the transition.
SOLOTONIC PROCCSSES
785
It may further be emphasised that non-linear excitations are of two main categories: the solitons proposed by Davydov and by Scott, representing local excitations serving to transport energy; and the type of excitation considered by Krumhansl and Alexander (1983) (as also in the present study) which are of the level change (kink) species which change the state of the system (here from the B to the A form of DNA) on its passage. The latter type of excitation (analogous to that obtained in a spin system through the propagation of the magnetic domain wall) belongs to the genre of ‘topological’ solitons (effecting systemic changes) and possess as such special features of stability preventing them from being damped out. The line of enquiry represented in this work may be considered to be a part of the programme undertaken by Krumhansl et al. (1985). 2. The Model.
Though the transition from the B to the A form of DNA is beset with formidable structural complexities, in the present level of description of macromolecular dynamics in an analytically tractable manner, it behoves us to attempt an isolation of dominant functionally-relevant modes. for which purpose the tilt angle & between the base normal and the helical axis (at the ith site) is chosen, since it represents the main stereochemical distinction and is the most labile element in the change. A further idealisation of the model is necessary whereby the site index is taken to a continuum described by the distance x along the chain, i.e. & + o(x). The rotational kinetic energy for the relevant motion of the base pairs is given by j$. where I is the corresponding moment of inertia per unit length. The potential energy must have two minima pertaining to the normal and tilted configurations, A#‘(+? - ~2)~/2, and a relative-humidity-dependent term. controlling the relative stability (as a function of water content, @=I? parameterised by v) of the two conformations. Lastly the system would militate against inhomogenity, providing a contribution $p(@/ax )2. representing the wall energy between regions of different conformation. Accordingly, the dynamics of the model DNA is governed by the Lagrangian density
yielding the equation of motion a24
I -at2
a24
'ax2
+ 2X@3- 3ahCJP+ (2X + v)@ = 0.
(2)
which. provided --Xa2 < v < 0. admits a solitonic solution (a discussion is provided in the Appendix)
786
r$ =
A. KHAN eta/.
l/ {+w/(XCY~ + v)}cosh
IJ
ha2 + v
(x - vt)
I(u2 - v2>
I
1
+ Xc2/(ha2 + v)
where v, the velocity of the soliton must be less than u = d/cc/I. On the other hand, at critical humidity (v = 0), when symmetric minima (of equal depth) occur at 4 = 0 and $I = (Y = 20”, which marks the onset of the desired transformation, a level-change soliton can be unleashed. described by $=-$tanhl($$r(x-vi)/-1].
which would travel along the DNA molecule tilting the base normal in its progress through the system. Considering the length of the molecule to be large in comparison with the wall thickness d=
u2 - v2
Xar2/41 ’ J or in other words neglecting end-effects, the energy of the soliton is given by
which for small velocities (v Q u) may be written as E = E. + irn*v’
where E,,, the o13d(h~)/6 and from which the conformational
+ . . .,
(4)
excitation energy for the solitary waves, is given by E, = the effective ‘mass’ of the soliton is FTZ*= CIor3~(X/~)?/6 threshold energy for the production of such a non-spreading change and the energy due to its motion may be found.
Some experimental results as well as con3. Discussiorz and Conclusions. formational calculations enable us to provide estimates of model parameters. The average length of DNA per base-pair is 1 = 3.6 X 1O-’ cm; the inclination of base normal to the helical axis for A DNA is & 2? 20” = n/9 radians (Dickerson et al., 1982); the moment of inertia per unit length has been estimated (Arnott and Hukins, 1973) to be / - 1.4 X 10-2serg/cm; the energy needed to shift a base-pair from one conformation to the other is expected (Englander et al.. 1980) to be of the order of 0.1 eV = 1.6 X 1O-l3 erg: the wall is expected to involve (Englander et al.. 1980; Jensen er al.. 1983) about - 10 base-pairs. the parameter X may be estimated from the curvature of the potential energy density [calculable from equation (1) to be ha2 I which according to the calculations of Zhurkin et al. (1978) using a modification of an algorithm due to Go and Scheraga (1970) is 3 X 10s3erg/cm lrad2. This yields the estimates h - 0.004 erg/cm. p - 1.3 X lo-“erg cm. consequently
SOLOTONIC PROCESSES
787
m* - 10-23g and E. - lo-r2 erg which meets the consistency check of having a wall involving - 10 base-pairs each involving -0.1 eV. This also provides an a posteriori justification for the use of the continuum approximation. Furthermore, if the velocity of the soliton is even as low as 0. lu it would take about a nanosecond to tilt the approx. 1600 base-pairs needed, for example, to initiate haemoglobic production, an estimate which is considered to be reasonable (Balanovski and Beaconsfield, 1982). Despite internal consistencies of the type of idealised models of DNA considered here, the question remains as to whether the actual DNA molecule can support such excitations and more importantly whether nature does in fact use this mechanism. It is, therefore, very important to carry out computer simulation studies on the dynamics of macromolecules, incorporating in a more tangible manner the main structural features of DNA and its various conformations in greater detail than an anlytically soluble model would permit. What is achieved by studies such as the present one is merely positing a prima facie case for solitons. However, as regards the search for solitons in situ, direct experimental verification may, it is feared, be very difficult and interpretations elusive. Nevertheless, it is necessary to pursue experiments to test these basic ideas. In the case of proteins there have been some claims (Webb, 1980), based on laser Raman spectra from metabolically active cells, for the existence of solitonic modes. Clearly careful experiments and cautious interpretations are called for. Studies on the kinetics of B to A transitions, and determination of the fraction transformed at critical humidity (which can provide a measure of the width of the soliton) need to be carried out. The authors are grateful to Drs Ashok Ranjan Thakur, Sunil Kumar Mukherjee and Probir Kumar Ghosh for several useful discussions, and also to the referee who brought to our attention the papers of Krumhansl and his group.
APPENDIX The idea of the solitary in August 1834, of:
wave owes its genesis to the observation
by John Scott Russell
a large solitary elevation, a rounded smooth and well-defined heap of water, which continued its course along the channel apparently without change of form or dimunition of speed. Subsequently, this phenomenon was demonstrated to find its explanation through a special solution of the Korteweg-De Vries equation for shallow water waves. Indeed various equations which are non-linear and also dispersive admit such solutions arising out of an exquisite interplay between these two tendencies acting in contradiction to prevent
788
A. KHAN et al
spreading of the waveform or packet, making possible relatively stable localised excitations and thereby providing interesting and efficient mechanisms for transport. These ideas have found extensive applications in systems as diverse as the plasma and elementary particles, solid state and non-linear optics. An excellent review of the subject is provided in an article by Scott et al. (1973) and in various books, e.g. Bullough and Candrey (1980). The equation with which we are concerned is of the form a2ti --U ar2
2 a% 2 + a@ + bq52 + c@j = 0.
It behoves us to perform a transformation of the moving frame by introducing the variable ,$’= x - vr, demanding at the same time that the solutions of interest (prapagating but non-spreading form-preserving excitations) should be functions of ,$’ alone. Accordingly the equation becomes
where the quantities with tildes-are the corresponding entities divided by (u2 - v’). In the problem under consideration in the present paper a = (a2X + v)/I, b = -3&h/I, c = 2x/I and u = dp/I. The nature of the solution depends on the relative disposition of these parameters. Though the search for solitonic solutions in general involves powerful techniques such as the inverse scattering method, for the present purpose it is very easily verified by direct substitution that the above equation possesses the desired solutions: 4 = [{&2b2
- 9a?)/18ii2)cosh{&(x
- vr)} - b/3ti]-‘,
In the regime ii > 0 and 2b2 > 9ir? which is realised in the case v < u and -ia2 for the problem at hand. Similarly I#I= (36/2b)[tanh{dd/4(x
- vr)} -
11,
obtains in the case ci > 0 and 26’ = 9ac which is realised for v < u and case of symmetric minima).
v =
0 (the special
LITERATURE Amott, S. and D. W. L. Hukins. 1973. “Refinement of the Structure of B-DNA and Implications for the Analysis of X-ray Diffraction Data from Fibres of Biopolymers.” J. molec. Biol. 81, 93-105. Balanovski, E. and P. Beaconsfield. 1982. “The Role of Non-linear Electric field Effects and Soliton Formation and Propagation in DNA Function.” Phys. Lerr. 93A, 52-54. Bhaumik, D., B. Dutta-Roy and A. Lahiri. 1982. “Solitary Waves and Macromolecular Systems.” Bull. math. Biol. 44, 705-713. Bullough, R. K. and P. J. Caudrey. 1980. Solirons. New York: Springer Verlag. Davydov, A. S. 1982. “Solitons in Quasi One-dimensional Molecular Structures.” Soviet Phys. Usp. 25, 898-9 18.
Dickerson, R. E., H. R. Drew, B. N. Conner, R. M. Wing, A. V. Fratini and M. L. Kopka. 1982. “The Anatomy of A-, B-, and Z-DNA.” Science 216,475-485. Englander, S. W., N. R. Kallenback, A.J. Heeger, J. A. Krumhansl and S. Litwin. 1980. “Nature of the Open State in Long Polynucleotide Double Helices: Possibility of Soliton Excitations.” Proc. narn. Acad. Sci. U.S.A. 77, 7222-7226. Frohlich, H. 1973. “Collective Behaviour of Non-linearly Coupled Oscillating Fields with Applications to Biological Systems.” Collect. Phenomena 1, 10 l-l 05,.
SOLOTONIC PROCESSES
789
Go, N. and H. A. Scheraga. 1970. “Ring Closure and Local Conformational Deformations of Chain Molecules.” Macromolecules 3, 178-l 87. Jensen, P., M. V. Jaric and K. H. Bannemann. 1983. “Soliton-like Processes During RightLeft Transition in DNA.” Phys. Let?. 95A, 204-208. Krumhansl, J. A. and D. M. Alexander. 1983. “Non-linear Dynamics and Conformational Excitations in Biomolecular Materials, Structure and Dynamics.” In NucZear Acids and Proteins (Eds E. Clemonti and R. H. Sarma), pp. 61-80. New York: Adenine Press. -, G. M. Wysin, D. M. Alexander, A. Garcia, P. S. Lomdahl and S. P. Layne. 1985. “Further Theoretical Studies of (Nonlinear) Conformational Motions in Double Helix DNA, Structure and Motions, Membranes.” In Nucleic Acids and Proteins (Eds E. Clemonti, G. Corongiu, M. H. Sarma and R. H. Sarma), pp. 407-415. New York: Adenine Press. Scot, A. C., F. Y. F. Chu and D. W. McLaughlin. 1973. “The Soliton: A New Concept in Applied Science.” Proc. IEEE 61, 1443-1483. Volkenstein, M. V. 1983. Biophysics. Moscow: MIR. Webb, S. J. 1980. “Laser-Raman Spectroscopy of Living Cells.” Phys. Rep. 60, 201224.
Zhurkin, V. B., Yu. P. Lysov and V. I. Ivanov. 1978. “Different Families of Doublestranded Conformations of DNA as Revealed by Computer Calculations.” Biopo/.vmers 17, 377-412. RECEIVED REVISED
6-30-84 7-23-85
ANGSHUTOSH KHAN Ashutosh College, Calcutta, India.
H DEBAJYOTI BHAUMIK Bose Institute, 93/l Acharya PrafuIIa Chandra Road, Calcutta-700009, India. n
BINAYAK DUTTA-ROY * Saha Institute of Nuclear Physics, 92 Acharya Prafulla Chandra Road, Calcutta-700009, India.
the conformations which the DNA molecule can adopt, the transition beween the A and B families, controlled by water content (relative humidity), seems to be implicated in the transcription process. Focusing on the main structural difference involved (tilting of base normals with respect to the helix axis), a model is constructed, solitary wave solutions of the resulting equation of motion are demonstrated and possible experimental implications indicated. Among
1. Zntroductiorl. After the double-helical structure of DNA was revealed in 1953 by Crick and Watson, subsequent studies of X-ray diffraction patterns from drawn fibres of DNA under diverse salt, alcohol and humidity conditions together with investigations on circular dichroism, infra-red spectrum and use of other solution techniques, led to the realization that the double-helical DNA structures can be categorised into two right-handed families-A and B-with fundamental differences as to the manner in which the base-pairs are stacked and the sugar-phosphate back-bone strands are wrapped along the helix axis. Furthermore. an unexpected left-handed Z-helix was found from X-ray analysis of single crystals. made possible by the development of tri-ester methods for the synthesis of pure DNA with a predetermined sequence. An excellent review of the anatomy of A-, B- and Z-DNA can be found in an article by Dickerson et al. (1982). wherein the salient structural distinctions are discussed. For some of the existing ideas *: To whom correspondence
should be addressed.
783
784
A. KHAN
etal.
and speculations regarding the functional roles of the different families of DNA conformations one may refer to the inexpensive textbook by Volkenstein (1983). Confining our attention, for the present purpose, to the A and B families, it may be discerned that the principal difference here resides in the inclination angle (4) of the base normals to the helix axis, being -20” and -0’ for the two forms, respectively, with a concommitant change in conformation of the sugar from a 3-endo to 2-endo, and a shift in the pitch. At low relative humidity (up to 70%) DNA crystallises in the monoclinic A form while at high humidity the hexagonal B form is obtained. The normal (or resting) state of DNA is presumably the B form and the mechanism and kinetics of its transformation to the A form is of considerable functional importance since this metamorphosis is very likely implicated in the process of transcription, whereby information stored in DNA through the sequence of bases, is conveyed to RNA (through the synthesis of RNA on the DNA template with the aid of RNAgpolymerase), which in turn transmits the genetic information to the ribosomes on which biosynthesis of proteins is accomplished. The propagation of localised excitations (or conformational changes) along macromolecules (or their aggregates) is often considered to play an important role in transport and transduction processes and in the kinetics of conformational transformations occurring in biology. The antagonists of this viewpoint, however, draw attention to the smallness of the lifetime of such excitations, which would indeed be the case if these were associated with local chemical groups, on the other hand stands the contention that the existence of collective modes with concommitant non-linearities in such systems makes possible relatively stable localised excitations through solitary wave solutions. Indeed Davydov (1982) has emphasised the possible role of solitons in a-helical protein molecules in their function, such as, for instance, their implication in the contractile activity of the actin-myosin complex in muscle tibres. The present authors too have sought (Bhaumik et al., 1982) for a solitonic excitation in macromolecular systems within the context of a model proposed by Frohlich (1973) involving polar modes and attendant elastic restoring forces. Englander et al. (1980) have predicted the possibility of a solitonic mode in the helix-coil transition involved in melting of DNA. Jensen et al. (1983) have invoked soliton-like processes during salt-induced right-left conformational changes in DNA, while Balanovski and Beaconsfield (1982) have considered, in like manner, the question of the A + B transformation. In the latter paper, however, attempts have not been made to categorise the structural features involved in the process. The object of the present study is to focus attention on the main structural distinction between A- and B-DNA and thereby to attempt a more concrete realisation of solitonic processes in the kinetics of the transition.
SOLOTONIC PROCCSSES
785
It may further be emphasised that non-linear excitations are of two main categories: the solitons proposed by Davydov and by Scott, representing local excitations serving to transport energy; and the type of excitation considered by Krumhansl and Alexander (1983) (as also in the present study) which are of the level change (kink) species which change the state of the system (here from the B to the A form of DNA) on its passage. The latter type of excitation (analogous to that obtained in a spin system through the propagation of the magnetic domain wall) belongs to the genre of ‘topological’ solitons (effecting systemic changes) and possess as such special features of stability preventing them from being damped out. The line of enquiry represented in this work may be considered to be a part of the programme undertaken by Krumhansl et al. (1985). 2. The Model.
Though the transition from the B to the A form of DNA is beset with formidable structural complexities, in the present level of description of macromolecular dynamics in an analytically tractable manner, it behoves us to attempt an isolation of dominant functionally-relevant modes. for which purpose the tilt angle & between the base normal and the helical axis (at the ith site) is chosen, since it represents the main stereochemical distinction and is the most labile element in the change. A further idealisation of the model is necessary whereby the site index is taken to a continuum described by the distance x along the chain, i.e. & + o(x). The rotational kinetic energy for the relevant motion of the base pairs is given by j$. where I is the corresponding moment of inertia per unit length. The potential energy must have two minima pertaining to the normal and tilted configurations, A#‘(+? - ~2)~/2, and a relative-humidity-dependent term. controlling the relative stability (as a function of water content, @=I? parameterised by v) of the two conformations. Lastly the system would militate against inhomogenity, providing a contribution $p(@/ax )2. representing the wall energy between regions of different conformation. Accordingly, the dynamics of the model DNA is governed by the Lagrangian density
yielding the equation of motion a24
I -at2
a24
'ax2
+ 2X@3- 3ahCJP+ (2X + v)@ = 0.
(2)
which. provided --Xa2 < v < 0. admits a solitonic solution (a discussion is provided in the Appendix)
786
r$ =
A. KHAN eta/.
l/ {+w/(XCY~ + v)}cosh
IJ
ha2 + v
(x - vt)
I(u2 - v2>
I
1
+ Xc2/(ha2 + v)
where v, the velocity of the soliton must be less than u = d/cc/I. On the other hand, at critical humidity (v = 0), when symmetric minima (of equal depth) occur at 4 = 0 and $I = (Y = 20”, which marks the onset of the desired transformation, a level-change soliton can be unleashed. described by $=-$tanhl($$r(x-vi)/-1].
which would travel along the DNA molecule tilting the base normal in its progress through the system. Considering the length of the molecule to be large in comparison with the wall thickness d=
u2 - v2
Xar2/41 ’ J or in other words neglecting end-effects, the energy of the soliton is given by
which for small velocities (v Q u) may be written as E = E. + irn*v’
where E,,, the o13d(h~)/6 and from which the conformational
+ . . .,
(4)
excitation energy for the solitary waves, is given by E, = the effective ‘mass’ of the soliton is FTZ*= CIor3~(X/~)?/6 threshold energy for the production of such a non-spreading change and the energy due to its motion may be found.
Some experimental results as well as con3. Discussiorz and Conclusions. formational calculations enable us to provide estimates of model parameters. The average length of DNA per base-pair is 1 = 3.6 X 1O-’ cm; the inclination of base normal to the helical axis for A DNA is & 2? 20” = n/9 radians (Dickerson et al., 1982); the moment of inertia per unit length has been estimated (Arnott and Hukins, 1973) to be / - 1.4 X 10-2serg/cm; the energy needed to shift a base-pair from one conformation to the other is expected (Englander et al.. 1980) to be of the order of 0.1 eV = 1.6 X 1O-l3 erg: the wall is expected to involve (Englander et al.. 1980; Jensen er al.. 1983) about - 10 base-pairs. the parameter X may be estimated from the curvature of the potential energy density [calculable from equation (1) to be ha2 I which according to the calculations of Zhurkin et al. (1978) using a modification of an algorithm due to Go and Scheraga (1970) is 3 X 10s3erg/cm lrad2. This yields the estimates h - 0.004 erg/cm. p - 1.3 X lo-“erg cm. consequently
SOLOTONIC PROCESSES
787
m* - 10-23g and E. - lo-r2 erg which meets the consistency check of having a wall involving - 10 base-pairs each involving -0.1 eV. This also provides an a posteriori justification for the use of the continuum approximation. Furthermore, if the velocity of the soliton is even as low as 0. lu it would take about a nanosecond to tilt the approx. 1600 base-pairs needed, for example, to initiate haemoglobic production, an estimate which is considered to be reasonable (Balanovski and Beaconsfield, 1982). Despite internal consistencies of the type of idealised models of DNA considered here, the question remains as to whether the actual DNA molecule can support such excitations and more importantly whether nature does in fact use this mechanism. It is, therefore, very important to carry out computer simulation studies on the dynamics of macromolecules, incorporating in a more tangible manner the main structural features of DNA and its various conformations in greater detail than an anlytically soluble model would permit. What is achieved by studies such as the present one is merely positing a prima facie case for solitons. However, as regards the search for solitons in situ, direct experimental verification may, it is feared, be very difficult and interpretations elusive. Nevertheless, it is necessary to pursue experiments to test these basic ideas. In the case of proteins there have been some claims (Webb, 1980), based on laser Raman spectra from metabolically active cells, for the existence of solitonic modes. Clearly careful experiments and cautious interpretations are called for. Studies on the kinetics of B to A transitions, and determination of the fraction transformed at critical humidity (which can provide a measure of the width of the soliton) need to be carried out. The authors are grateful to Drs Ashok Ranjan Thakur, Sunil Kumar Mukherjee and Probir Kumar Ghosh for several useful discussions, and also to the referee who brought to our attention the papers of Krumhansl and his group.
APPENDIX The idea of the solitary in August 1834, of:
wave owes its genesis to the observation
by John Scott Russell
a large solitary elevation, a rounded smooth and well-defined heap of water, which continued its course along the channel apparently without change of form or dimunition of speed. Subsequently, this phenomenon was demonstrated to find its explanation through a special solution of the Korteweg-De Vries equation for shallow water waves. Indeed various equations which are non-linear and also dispersive admit such solutions arising out of an exquisite interplay between these two tendencies acting in contradiction to prevent
788
A. KHAN et al
spreading of the waveform or packet, making possible relatively stable localised excitations and thereby providing interesting and efficient mechanisms for transport. These ideas have found extensive applications in systems as diverse as the plasma and elementary particles, solid state and non-linear optics. An excellent review of the subject is provided in an article by Scott et al. (1973) and in various books, e.g. Bullough and Candrey (1980). The equation with which we are concerned is of the form a2ti --U ar2
2 a% 2 + a@ + bq52 + c@j = 0.
It behoves us to perform a transformation of the moving frame by introducing the variable ,$’= x - vr, demanding at the same time that the solutions of interest (prapagating but non-spreading form-preserving excitations) should be functions of ,$’ alone. Accordingly the equation becomes
where the quantities with tildes-are the corresponding entities divided by (u2 - v’). In the problem under consideration in the present paper a = (a2X + v)/I, b = -3&h/I, c = 2x/I and u = dp/I. The nature of the solution depends on the relative disposition of these parameters. Though the search for solitonic solutions in general involves powerful techniques such as the inverse scattering method, for the present purpose it is very easily verified by direct substitution that the above equation possesses the desired solutions: 4 = [{&2b2
- 9a?)/18ii2)cosh{&(x
- vr)} - b/3ti]-‘,
In the regime ii > 0 and 2b2 > 9ir? which is realised in the case v < u and -ia2 for the problem at hand. Similarly I#I= (36/2b)[tanh{dd/4(x
- vr)} -
11,
obtains in the case ci > 0 and 26’ = 9ac which is realised for v < u and case of symmetric minima).
v =
0 (the special
LITERATURE Amott, S. and D. W. L. Hukins. 1973. “Refinement of the Structure of B-DNA and Implications for the Analysis of X-ray Diffraction Data from Fibres of Biopolymers.” J. molec. Biol. 81, 93-105. Balanovski, E. and P. Beaconsfield. 1982. “The Role of Non-linear Electric field Effects and Soliton Formation and Propagation in DNA Function.” Phys. Lerr. 93A, 52-54. Bhaumik, D., B. Dutta-Roy and A. Lahiri. 1982. “Solitary Waves and Macromolecular Systems.” Bull. math. Biol. 44, 705-713. Bullough, R. K. and P. J. Caudrey. 1980. Solirons. New York: Springer Verlag. Davydov, A. S. 1982. “Solitons in Quasi One-dimensional Molecular Structures.” Soviet Phys. Usp. 25, 898-9 18.
Dickerson, R. E., H. R. Drew, B. N. Conner, R. M. Wing, A. V. Fratini and M. L. Kopka. 1982. “The Anatomy of A-, B-, and Z-DNA.” Science 216,475-485. Englander, S. W., N. R. Kallenback, A.J. Heeger, J. A. Krumhansl and S. Litwin. 1980. “Nature of the Open State in Long Polynucleotide Double Helices: Possibility of Soliton Excitations.” Proc. narn. Acad. Sci. U.S.A. 77, 7222-7226. Frohlich, H. 1973. “Collective Behaviour of Non-linearly Coupled Oscillating Fields with Applications to Biological Systems.” Collect. Phenomena 1, 10 l-l 05,.
SOLOTONIC PROCESSES
789
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