SOLITON STATISTICAL MECHANICS AND THE THERMALISATION OF BIOLOGICAL SOLITONS R. Bullough, D. Pilling, Yi Cheng, Yu-Zhong Chen, J. Timonen
To cite this version:
R. Bullough, D. Pilling, Yi Cheng, Yu-Zhong Chen, J. Timonen. SOLITON STATISTICAL ME- CHANICS AND THE THERMALISATION OF BIOLOGICAL SOLITONS. Journal de Physique Colloques, 1989, 50 (C3), pp.C3-41-C3-51. 10.1051/jphyscol:1989306. jpa-00229445
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SOLITON STATISTICAL MECHANICS AND THE THERMALISATION OF BIOLOGICAL SOLITONS
R.K. BULLOUGH, D.J. PILLING, YI CHENG, W-ZHONG CHEN and J. TIMONEN*
Department of Mathematics, ,UMIST, PO Box 88, GB-Manchester, M60 1120, Great-Britain 'Department of Physics, University of Jyvdskyld, SF-40100, ~yvdskyld, Fin1 and
Abstract - The calculation of the equilibrium free energy of integrable models like the sine-Gordon and attractive nonlinear Schrodinger models is discussed in the context. of biological molecules like DNA: the thermalisation process (approach to equilibrium) is also discussed. The sine-Gordon model has a "repulsive" form which is the sinh-Gordon model. The approach to equilibrium of the sinh-Gordon model is described in all completeness in terms of a quantum mechanical master equation at finite temperatures. Although the dynamical evolution of the master equation as written is a solved problem, only the equilibrium solution is examined in this paper. The equilibrium free energy is calculated exactly as an integral equation for certain excitation energies. at finite temperatures. Bose-fermi equivalent forms of this integral equation are given. The hose form yields a similar integral equation in classical limit. The iteration of this yields a low temperature asymptotic series for the classical free energy which checks against the result of the transfer integral method (TIM). Results for the zero temperature quantum eigenenergies are found. A further discussion of the dynamics of the approach to thermal equilbriurn is made.
1~- INTRODUCTION
It has been suggested (eg./1,2,3/) that the classical sine Gordon model
ax, - @tt = m2 sin O (1)
is a good model to describe soliton excitations on DNA (here O means a2Q/ax2, etc, the left side is a,, - co-20tt, m is a mass or the wave number mcoft-i~and units are chosen so A = co = 1). Likewise the model may have some relevance to the transmission of soliton-like excitations on the protein a-helix. But here the non-relativistic form of the s-G model which is the non-linear Schrodinger model
seems preferable /4,5,6,7/(Davydov's mode1/4,5/ coincides with (2) only in that his soliton is the soliton solution of (2)). Note that the field Q in (1) is real. But in the NLS model (2) Q is complex. Moreover there are actually two NLS models: c is a coupling constant in (2) and c
In this paper we focus attention on the sine-Gordon model (s-G), equation (1) and its "repulsive" counterpart the sinh-G model which has no solitons. Both are integrable models /8,9/. The soliton solutions of the s-G are the well-known kink and antikink solutions
The kink solution (tve sign in (3)) takes @ from zero to 217 as x goes from -oo to +m: the antikink takes O from 2n ko zero. Thus the kink (3) carries a twist of 2n up x with speed V: the antikink (3) carries a twist of -271.
The point of such solutions in the biological context is that
ax = *2rn(l-~~)-~sech (+rn(x-vt)(l-~Z)-%] ,
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1989306 C3-42 JOURNAL DE PHYSIQUE
These are narrow soliton pulses of width - (1-V2)* m-I. The velocity V is scaled against co, so as the speed approaches cop the sound velocity ie. V+1, these solitons become very narrow and have large amplitudes. Both kinks and antikinks carry energy ~(1-v2)-%.If we define a momentum pk = MV(~-V~)-* this energy is
We encounter this later. The number M is the kink mass: M = 8mya-l in which yo>O is the coupling constant of s-G. We remark further on yo shortly.
These remarks are intended to show that the s-G solitons (kink or antikink) are essentially compact localised pulses carrying specific energies (5). In the case of S-GI these solitons are topological solitons and also carry twists of 2n(-2n). Both the pulses, and their twists, can be viewed as "bits" of information. The pulse as a "bit" has been used in a "shift register" with pico-second access /lo/. The soliton solutions of the NLS, equation (2), with c Neither (1) nor (4) shows evidence of damping. But it is easy to show that even when the s-G system is damped, the solitons remain acceptable solutions (cf. eg. /13/). It is these several facts that could make the soliton an important mode of energy and information transport in biological systems. Certainly the s-G (1) plays an important role in nonlinear physics /14/. For example it describes the excitations of ferromagnetics like CsNiFB /15,16,17/ and antiferromagnetics like TMMC /16,17/. These excitiations are present in thermal equilibrium and apparently govern the neutron scattering cross-sections /16,18/. In biological molecules, the soliton must be subject to thermal agitation. To describe this one can introduce a random force F(t) into equations of motion like (1). This causes diffusion and damping so additionally the left side of (1) gains a term in -K@t associated with the -@tt:K is a damping constant. The random force F(t) appears in a quantum theory developed in Heisenberg representation. Another way to proceed is to work with a quantum theoretical master equation in Schrodinger representation /19/. We do this briefly in this paper (in 5 4). A steady state solution of the master equation is (exp-PH)Z-l: E1 is the temperature T (Boltzmann's constant kg = l), H is the Hamiltonian (operator) and Z is the partition function. The free energy F = -K1ln Z. We can compute Z by the methods of statistical mechanics. It is the statistical mechanics of integrable models like the s-G (I), or the NLS models (2), with which this paper is primarily concerned. To this end we need the complete solution of s-G. In addition to the kinks and anti-kinks there are bound pairs of these, called "breathers" namely @(x,t) = 4 tan-I [tan u sin eI sech +] % = (m sin p)(x-vt)(l-V2)-% eI = (m cos &f)(t-VX) (1-v2)-*. These have energies The rest energy is Mb = 2M sin u. Since 0 < i.d < Hrr, the breather masses form a band in 0 < Mb < 2M. Since m cos u is a frequency (in the chosen units) the breather solution (6) has an internal oscillation sin eI modulated by a sech envelope all under the tan-I function. For small enough !.I d~ (x,t) - 4mi.d sin (m(t-Vx) (I-V~)-~) (8) a harmonic solution of very small amplitude. This is actually a solution of the linearised s-G, the Klein-Gordon (KG) equation However it is well known /9/ that the s-G itself has 3 sorts of solution : the kinks (and antikinks) (3), the breathers (6) and "radiation" approximately described by (8). We shall call this radiation "phonons" in this paper. Thus: the complete solution of s-G (under vanishing boundary conditions at x = w /8,9/)is made up of kinks plus antikinks plus breathers plus phonons. However it is already clear that it may not be very easy to distinguish phonons from small amplitude breathers. This fact plays a role in the statistical mechanics (5 3). The coupling'constant yo introduced by equation (5) allows a continuation from Yo + -Yo. If we set d + eOd(which is actually a canonical transformation /8/) the s-G (1) is Then, as yo -t -Yo, (10) becomes the sinh-G in JE a. Moreover as yo + 0, (10) becomes the KG (9). Thus sinh-G is obtained by continuation in yo from s-G; and both s-G and sinh-G contain KG as yo + 0. Because sinh-G and repulsive NLS have no soliton solutions they might seem to be uninteresting. This is not the case. Certainly the statistical mechanics of these two models is easier than that of s-G or attractive NLS. It is for this reason we shall mostly be concerned with the statistical mechanics of sinh-G in this paper 2 - THE DRESSED AND UNDRESSED NUMBER DENSITITES The number of classical or quantum solitons of the s-G excited at temperature B1 in thermal equilibrium is of physical interest. If one computes the 1-particle partition function m +H L --Q) -H L for kinks and uses nk = L-I {a(B1 In Z)/au],,=o (D is here a chemical potential) one finds /20/ The series on the right is an asymptotic expansion valid for small (MP)-I (low T). From ZE for antikinks one finds nk = ny; while the breather density is similarly /20/ r 1 In (12) and (13) both Kn and In are modified Bessel functions as in 1203. One finds the free energy per unit length as and FKG is the free KG contribution FKG = lim kla-1 ['ln(n&-I) + H ma - 11. (15) a+O It is no surprise that (15) diverges: the KG, equation (9), is in effect, a bunch of harmonic oscillators with dispersion (Fourier transform (9) to k-space). Classically the divergence (15) is then the classical ultra-violet divergence. This does not arise as such in the quantum case (R. K. Bullough, Yu-zhong Chen, and J. Timonen, in preparation): this is because FKG arises from the phonons and there are no phonons in the quantum case of s-G (/21/ and see below). From (12), (13), j14), (15) we thus have C3-44 JOURNAL DE PHYSIQUE with FKG given by (15). Although (15) diverges one finds the form (15) with a>O comes from the KG (9) alone by using the cut-off Jk( plus terms in e-213M, e-3m, etc. multiplying further asymptotic series. The extra factor 2(mP), the coefficient -3% in the kink-antikink series, and the term in (MP)-I in the breather series in (18) are due to "dressing" effects from the phonons in the problem: the new series multiplied by e-Zm, are "multisoliton" (multikink or multiantikink) effects. An important conclusion emerges from the comparision of (17) and (18). Suppose the s-G kink is a good model of a biological soliton on DNA (say), /1,2,3/: it will be subject to thermal agitation in vivo and, in the absence of other forces, a random force F(t) describing its Brownian motion under this agitation would ultimately drive the soliton into thermal equilibrium with a free energy FL-* = -nkrl with nk given by (12). However, as such solitons accumulate, they must eventually thermalise still further reaching a total free energy density (18) in final thermal equilibrium. Of course it is not clear that the thermalisation sequence necessarily passes through (17). But certainly (18) is the equilibrium free energy while -Pnk, nk from (12). is the equilibrium for a single kink undergoing Brownian motion. The dressing process to (18) will take time and a kink originally travelling with energy (4) >> P-I will slow down on some time scale K~-~say (Kk is a damping constant for this dressing process). To calculate ~k we need to describe this whole thermalisation process much more completely. Such a dynamical calculation is still to be done but we sketch some ideas in this connection in 54 next. The conclusions so far are:- (1) In the absence of other dynamical forces, a biological soliton (kink), will thermalize on some time scale K~-~eventually reaching thermal equilibrium; (2) In thermal equilibrium the number densities and free energies of classical kinks and antikinks are strongly dressed by the phonons (although l3-I - 300'K we can expect mP is small so FL-I for kinks in (18) is increased by this dressing process); (3) The number densities and free energies of the breathers are also substantially changed: we can show that the classical breathers actually become large amplitude phonons /21/: these are interacting phonons which dress the kinks and antikinks: they give rise to the phonon series in (18) while classical breathers actually disappear /21/. (4) Free solitons (kinks, antikinks) thermalise to dressed solitons in equilibrium: free breathers thermalise to phonons and otherwise disappear. (5) These remarks refer to classical kinks and antikinks and breathers: the quantum s-G behaves quite differently (work by the authors to be reported). 3 - THE ROlaE OF THE PHONONS To assert as we have done that the series in (18) is not the breather series of (17) at all may seem presumptuous. However, we know from the TIM that the free energy of sinh-G is /23/ (this checks with (18): put yo -r -yo bearing in mind there are no soliton solutions of sinh-G). We also know /24/ that sinh-G has only phonons (and no kinks, antikinks or brehthers). Thus the identification of the series in (18) as a phonon series rather than a dressed form of the breather series of (17) is really that the series for sinh-G in (19) is derived from the sinh-G phonons; then the series in (18) follows by yo-' -yo reaching s-G. The situation is the more extraordinary in that as integrable models both sinh-G and s-G are completely integrable /8,9/ with action-angle variables /8,9/. In s-G in particular the Hamiltonian H[pl in these variables is /8,9/ m - (20) In this Hamiltonian M is the kink mass as earlier and pi, pj are action variables (compare the expression (5) for the kink or antikink energies). Likewise 59 and 89 are action variables (compare (7)) and 89, 0 ( 89 < Mr, is the action variable for the internal degree of freedom of the breather. The final integral is the phonons' contribution: w(k) is given by (16), the P(k) are action variables and they have canomical Q(k), 0 ( Q(k) < 277, angle variables such that the Poisson bracket {P(k), Q(kJ)}= S(k-k') /8,9/. Now since sinh-G has neither kink, antikink or breather excitations its comparable H[p] must be (it is /8/). The only problem with this (apparently) is that (21), found however by Fourier transform not the spectral transform /8,9/, is the H[p] for linear KG (9): it has exactly the same form. The situation is resolved in the following way /8,23/. Although sinh-G has no solitons and no breathers its thermodynamics is determined by large amplitude interacting phonons in thermal equilibrium: they interact through pair-wise phase shifts, and this statement applies to both the quantum sinh-G and the classical sinh-G /8,23,24/. The case of s-G is more bizarre: first in the quantum case there are no phonons /21,24/, only quantised kinks, antikinks and breathers; second in the classical case /21,22/ there are no breathers, but there are kinks, antikinks and classical phonons; third the classical limit of the quantum case of s-G /25/ shows that the quantum breathers become large amplitude classical phonons interacting through phase shifts exactly as in the case of sinh-G described already. The authors are slowly publishing the details of these (surely remarkable) features of the thermodynamics of the s-G model. 4 - THE SINH-G MODEL IN A HEAT BATH Although the dynamics of a single soliton of the a-G travelling at speed V and thermalisins to its equilibrium velocity in some time K~-~would be the ultimate aim of the work sketched in this paper, this calculation is not yet achieved. There is a substantial simplication if one uses the sinh-G model instead of the s-G (though there are now no kink or antikink solutions). The simplicity stems from (21) which shows sinh-G is a bunch of (classical) oscillators. A bunch of quantum oscillators in a heat bath at temperature T = TJ-l > 0 satisfies the master equation /19/ C3-46 JOURNAL DE PHYSIQUE and [ak, akt] = S(k - k') : y(k) are rate constants for each k. Note that (i) for bose oscillators this master equation (22) is a solved eguation, eg. in equilibrium (ii) aktak o P(k) (particle number) ; (iii) a steady state solution is p = e-rn / Tr e-rn (which follows from R/(1 + R) = e-m(k)). Here and in the master equation (22) H is the Hamiltonian operator in Schradinger representation: p is the density matrix. For a steady state solution, which is all we can compute in this paper, one should as expected compute the partition function Z = Tr e-BH. But, with H as given below (22), The classical limit of this is m and this diverges. However, if we use the cut-off 0 < Ikl < na-l, What has gone wrong? What has gone wrong is that we have failed to address the problem of the thermodynamic limit. 5 - TIIE THERMODYNAMIC LIMIT We must reach a finite density thermodynamic limit - achieved typically by using periodic boundary conditions of finite period L (say) such that for N particles in L /8,21/ lim N=?i>O L- L For KG this has no special consequences (FL-I is still given by FKG) but for (nonlinear) sinh-G the effect is dramatic: we find wh_ere_Pn= O(1): Pn <->P(Ti)dk (L+ -)and this implies P(E) is O(L): then L-IPn = L-t P(E)dg = e(k)dk (say) and p(E) is a finite densitiy. This explaiss the phrase, "large amplitude" phonons used earlier: P(k) -t m. Note that (27) yields Iw(k) p(k)dk as L + - . d :-Y y0m2 [Xu(*')-k9w(k)]-I (Ac is the classical phonon;phonon phase shift and the "interaction" is contained through (28) where the allowed modes k, are solutions of the system (28); kn r 2nn L-I, the free field Note that, for KG, yo = 0 and kn = kn in (28). One needs to know too that (28) applies in the quantum case also /23/. But then A, + Ab (see next) and P, in (28) has Pm = 0, 1, 2, ... as for bose quantum oscillators. There are actually two cases (at least) /8,23/: P, = 0, 1, 2,... for bosons with Ac + Ab and Pm = 0, 1 for ferrnions with Ac + Af. One can choose either case for the quantum theory and results are identical! We find /23/ with .Yo" = yo/(l + Yo/8n) and the smooth branch -2n ( 4 ( 0 must be taken for the tan-1 in the fermion case. Then Ab(k, kt) = Af(k, k') + 2n 8(kP-k) where 8(k) = 1, k > 0, 8(k) = 0, k < 0. 6 - THE FREE ENERGY The procedure now is to calculate an entropy S and so the free energy F: this procedure substantially generalises the work of Yang and Yang on the repulsive NLS /8,26/ - namely to a boson description and to the classical case also (we called it method of 'generalised Bethe ansatz' /8,22,23/). A fundamental set of modes are the G: set k a function of the G (ie. k, + k = h(g)). /27/ilI, Then as L + w (28) is (now call B + k) J --m The allowed modes define a density of allowed states f(E) ( or f(k)) (say) such that so (30) means (using bosons) (lj~eferenceto the Ref. /27/ shows there is_ a mistake in the argument-of that paper, where, as a result of a copying error, k, and kn are confused (roughly kn and kn are to be interchanged in the 53 of /27/). A result is our (32) and (44) below gain a wrong sign. One also has the energy E (Hamiltonian), momentum P, and number of particles (bosons) 0 w The entropy (for bosons) is computed from the number of possible states in dk which is [L(p + f) dkl!/[L p dkl! [L f dkl! (34) The entropy per unit length is JOURNAL DE PHYSIQUE s L -'= I C(f + p) ln(f + p) - f inf - p in pldk -m (p is now the boson density not the density operator of (22)) and we minimise FL-I = (E - B-l~)~-lie. we set S(FL-')/S~= 0. We readily find m --Q3 (35) If we define (f + p)p-' t exp *(k) we find the energies ~(k)satisfy and we can go on to show that a We have thus regained (23) with however the condition that the energies ~(k)satisfy (36)! The classical limit of (37) with (36) puts In(1 - R-E(~))+ ln PE(k), and Ab + & as in (28), (this is the small yo "limit" of (29) in the boson form: there is now a singularity at k = k', which not true of +). The iteration of this classical limit of (37) with (36) yields the asymptotic expansion of (19) exactly /23/. This explains how this result is due to large amplitude phonons (thermodynamic limit L-lP, + p(k)dk, p(k) finite) interacting through phase shifts (the Ac). One can either repeat the calculation for fermions instead of bosons (Pm = 0, 1: Ac + 4) or transform (36) with (37) - demonstrating strict equivalence. For the latter 4 = Ab - 2n 8 - while new energies Z(k) are defined by ln(1 + e-@(k) ) = Iln(1 - e-@(k) ) . For the former it is convenient in any case to introduce: a chemical potential LI by minimising the negative pressure -p rn FL-1 - LINL-~ = (E - B'S - pN)~-l. The number of possible states (34) is not changed but now Either way the energies prove to safisfy (for finite u) while There are two interesting free particle limits: both arise where sin(%yow)+ 0, but this is where yo/(l + yo/8v) = 0 or 8n, so yo = 0 or yo-. In both cases r(k) = ~(k)-p,just ae for a gas of free bosons: since yO+O is the free KG, this is expected but the result when Yo@ is not expected. In fact the repulsive NLS (2) (c>O) has the same integral equations like (39) with (40) in fermion form except that ~(k)= k2 and Ap = -2tan-l(c/(k-k')) (smooth branch) /26/. When the coupling constant c-, ~(k)= k2-u, a gas of free fermions. This is the impenetrable bose gas /28/. 7 - THE QUANTUM MECHANICS From the results of 56 we we can obtain eigenenergies at T=O (ie. the quantum mechanics). We look for the aeros at k= *kf of &(ak)=O. Then &uo is fixed by this kf and E(k) > 0, Ikl > kf; z(k) < 0, Ikl < kf. Then from (39) as B1 = T + 0 In analogy with the repulsive NLS /27,29/ energies El I E-Eo are excitation energies above the ground state energy Eo while El = E (k*) - c (kh). (43) El corresponds to a fermion (in fermion description) in a state kp > kf together with a hole in a state kh < kf. The energies t(kp) and E(kh) are solutions of (42). To get the ground state energy Eo, and 80 from (43) the excitation energy E = El + %, use ~(k)= 0, IkJ > kf and f(k) = density of (fermion) states = 8 (k). Then ~(k)becomes a solution of (32) in the form +K f while -kf In this description p(k), originally a density of bosons per unit length, must be interpreted as a density of fermions per unit length. This is precisely what is found by an ab initio fermion description (the error in reaching (44) from the boson description is that something like a Bose-Einstein condensation occurs and the passage to (44) at T = 0 needs seperate analysis (this analysis is not yet wholly done)). 8 - FURTHER AND FINAL REMARKS In the case of the s-G it is not yet clear how to couple quantum solitons and breathers (the only quantum excitations) to a heat bath as was done for the phonons of sinh-G in 54. Note again that for the full quantum s-G there are only solitons (kinks and antikinks) and quantum breathers. Still the equilibrium analysis in terms of fermions can still be done: it yields a system of n-1 integral equations for distinct fermion energies: n = [8n yo-'] = integral part, there are n-2 quantum breathers and one kink-antikink of double weight (see eg. Ref./25/ and references as well as work by the authors to be published). The quantum system has a semi-classical limit /25/: this can also be derived /22/ from an H[p] Compared to (20) this neglects breathers: compared with (21) it will add kinks and antikinks (fermions) to phonons (bosons). It, nray be possiblc to write a quantum operator form for (46) which extends (22) to this case: if (22) is formally unchanged the damping of the solitons (kink and antikink) takes place only through interaction with the phonons which alone are coupled to the heat bath. The stability of the solitons perhaps makes such a model a good approximation. Evidently a dynamical study of the thermalisation of s-G solitons will require much more work: it is necessary to choose the phonon damping constants y(k) to model their thermalisation adequately. Then it is necessary (in terms of the model just described) to calculate the decay time K~-~of a kink in terms of these ~(k). C3-50 JOURNAL DE PHYSIQUE One of us (RKB) has already speculated /30/ on the calculation of Kk somewhat in the terms reported here. But it will be clear from reference to /30/ that we have advanced considerably in our understanding of the equilibrium steady state of both sinh-G and s-G, quantum or classical, since that paper. Ref./30/ also compares with the work of Wada and Schrieffer /31/ who compute a diffusion constant D = 0.516 woa2 (kgT/mOo2 wo2))' for the 0-four model (c = velocity) which though not integrable has for B>O long lived kinks and antikinks: Qo = t ((AI/B)~is a zero of the right side. By expanding Q about Qo there are "phonons" with dispersion wZ = 21AIm-1 + cZkZand wo = (21~lm-~))I.In D, kgT = k1 while a is a lattice spacing much as was used for ZKG earlier. To reach their result for D, the authors use finite amplitude phonons and work to second order: we here use pn for these amplitudes and following /31/, since < pnz > - kBT (= Pi) (<...> means thermal average), < P,'> - (kB~):to second order. Then D proves to be D - (kBT)Z. Now the analysis to the free energy FL- of this paper also uses finite amplitude phonons as explained, and the correction term to HL-I = L-I Dd(kn) Pn introduced through (28) is of the order Pn2: since lpnI2 = Pn, D-(~BT)~. However this result tells us only that there is a damping term perhaps like -Kka@/at to be included on the left side of the s-G equation (1). In Einstein's theory of Brownian motion for simple particles of mass mo and position x, D and K are related by mox -, mox + m0Kx with < xZ> = 2kBTt/~mo= 2Dt. More typically mo K = kgT mo/~mot where mot is an effective mass. This theory cannot apply directly if D-(~BT)~since then K increases as T+O. Still the fluctuation dissipation theorem tells us that if -K &/at (= -KQ~) adds to the left side of the s-G equation (I), then we expect D = f(T)~-lwhere f(T) is a function of temperature to be determined. These rather imprecise considerations suggest again that (1) is to be replaced by @ti+ K@t = qx- m2 sin 6 + F(t) where K-~is an estimate of the time K~-~for a (biological) soliton of s-G type to reach equilibrium. We stress that all of the quantum and classical equilibrium results for FL-I found as in 56 are exact /8,23/: they apply to any integrable model /8/ whose classical H[p] takes the form (21) in action variables (with appropriate w(k) and A), and there are very many such systems. We stress too that the comparable results (eg. /21,22,32/ and work by the authors to be published) for integrable systems like s-G which classically have soliton solutions are similarly exact. We have exact results of the same sort for the classical Landau-Lifshitz model and the Toda lattice (Yu-zhong Chen, Ph.D thesis, U. of Manchester to be submitted). The results for the Toda lattice are relevant to the paper on solitons in DNA presented at this meeting /33/. In contrast with these exact equilibrium results we have still to provide an adequate analysis of the dynamical approach to equilibrium - though for sigh-G for example this is apparently provided by (22) with allowed modes k restricted to the k satisfying the quantum bose form of (28). Even so we must acknowledge that though the equilibrium theory is in good shape there is still much to be done before we can adequately- describe the dynamics of solitons on realistic models of biological systems. REFERENCES Yomosa, S. Phys. Rev. A 27 (1983) 2120. Englander, S. W., Kallenbach, N. R., Heeger, A. J., Krumhansl, J. A. and Litwi, S., Proc. Nat. Acad. Sci. USA., 77, 7222. 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(World Scientific: Singapore, 1985) pp. 458-466. /31/ Wada, Y. and Schrieffer, J. R., Phys. Rev. B. 18 (1978) 3897. /32/ Timonen, J., Pilling, D. J., Chen, Yu-zhong, and Bullough, R. K. in "Plasma theory and nonlinear and turbulent processes in physics" V. G. Bar'yakhtar, Y. M. Chernousenko, N. S. Erokhin, A. G. Sitenko and V. E. Zakharov eds. (World Scientific: Singapore, 1988) pp. 878-896. /33/ Muto, V. "Thermally generated soliton in DNA". This meeting. Note in proof: see also the reference /34/ Muto, V., Halding, J., Christiansen, P.L., and Scott, A.C., J. Biomolecular Structure & Dynamics ISSN 0739-1102 5 (1988) 873. Additional note in proof: In the paragraph below that containing eqn. (18) we very tentatively suggest that there might be a thermalisation process in which the undressed kink density (12) is actually realized if only transiently (the final density is certainly that given in terms of free energy (18)). However, it is really pretty clear that even if the biological molecule were initially subject only to Einstein type random force F(t) of Brownian motion due to a surrounding solvent (say) then (12) is realizable only if the temperature is very low so that there are very few phonons present. As indicated below (18) solitons also dress the solitons (they provide the multisoliton terms adding to (18)). We are not yet in a position to say which dressing process dominates the thermalisation sequence.