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Solitary waves and macromolecular systems
Bulletin of Mathematical Biology, Vol. 44, No. 5, pp. 705-713, 1982 Printed in Great Britain.
0092-8240/82/050705-09503.0010 Pergamon Press Ltd. © 1982 Society for Mathematical Biology
SOLITARY WAVES AND MACROMOLECULAR SYSTEMS DEBAJYOTI BHAUMIK Bose Institute, 93/1 A c h a r y a Prafulla Chandra Road, Calcutta 700009, India BINAYAK DUTTA-ROY
Saha Institute of Nuclear Physics, 92 Acharya Prafulla Chandra Road, Calcutta 700009, India AvIJIT LAHIRI Vidyasagar (Evening) College,
Calcutta 700006, India
Macromolecules and their aggregates (such as protein bundles in biomembranes) possess polar modes which, when excited, tend to deform the system and call into play elastic restoring forces. A model of such systems, characterised typically by electric polarisation modes stabilised on the one hand by quartic self-interactions and on the other through coupling to the elastic deformations, admits the possibility of localised excitations (solitary waves) propagating with subsonic velocities, possessing the features of relative stability and efficient transport characteristics (associated with the collective nature of the phenomena), and at the same time provides a mechanism of control and variability which could be of considerable interest in biology.
Introduction. The propagation of localised excitations along protein molecules is often considered to play an important role in transport and transduction processes occurring in biology. The antagonists of this view-point, however, draw attention to the smallness of the life-times of such excitations, which would indeed be the case if these were associated with local chemical groups; on the other hand, it is our contention that the existence of collective modes in macromolecules (or their aggregates), with concommitant non-linearities makes possible relatively stable localised excitations, providing interesting mechanisms for transport. Indeed, the potentiality of solitary waves in adopting the role of such carriers has been emphasised by Davydov (1977). Nevertheless, the full richness of the underlying structure and the scope for control and influence by external factors, features of paramount importance in biology, has been left unexposed by that author. The object of the 705
706
D. BHAUMIK, B. DUTTA-ROY AND A. LAHIRI
present work is to uncover some of these aspects within the context of a model which has yielded some insight into the dynamics of macromolecules. In a series of papers, Fr6hlich (1968a, 1968b, 1968c, 1969, 1970, 1972, 1973, 1975a, 1975b) has conjectured that macromolecules often possess polar modes which, when excited, tend to deform the system in various ways, calling into play and counteracted by elastic forces which determine the shape of the unexcited system. The present authors (Bhaumik et al., 1976a, 1977, 1978a, 1978b) and others (Wu and Austin, 1977, 1978) have applied these ideas to discuss the possibility of a metastable 'ferro-electric' state with large dipole moment, consequent modification of the Van der Waals interaction between such molecules and implications on the mechanism of enzyme reactions. An appreciable body of experimental studies seem to suggest existence of collective polar modes of macromolecular systems and their role in biological phenomena. The extraordinary dielectric properties of biological materials (Fr6hlich, 1975b) emphasised by Fr6hlich, in particular the low lying vibrations in 1011 Hz frequency region (presumably with giant dipole moments), profound non-thermal effects (including marked enhancement of biological activity) by millimeter irradiation on biological samples (Webb and Booth, 1969; Deryatkov et al., 1974; Berteaud et al., 1975; Grundler et al., 1977) and results of laser raman spectroscopy (Webb and Stoneham, 1977) strongly suggest the existence and functional importance of collective polar modes of macromolecules. In our previous investigations attention was confined to the stationary configurations of the molecules, while this study is devoted to dynamical behaviour, particularly to the movement through the macromolecule of localised excitations.
The Model. The model (Bhaumik et al., 1977) is defined in terms of polarisation and acoustic fields P(x, t) and A(x, t), which are functions of time t and position x in the macromolecule (for simplicity, the coordinate along the chain has been taken as the only relevant dimension, and as vibrations in large molecules are frequently isotropic, the vectorial character of the fields has been suppressed). No doubt this description mutilates the detail of the chemical and stereo-structure of the molecule, but it provides a description which brings into focus the role of collective modes in the dynamical behaviour of the system, which is thus taken to be governed by the Lagrangian density 21[~2 + =
A 2 (c~P~ 2 _ o "2 (~A~ 2 1 2 2 - T
T
\ ax /
-
O oI. -
c (c~A~2p2
1 d2p4" --g
(1)
SOLITARY WAVES AND MACROMOLECULAR SYSTEMS
707
The first two terms represent the kinetic energy densities, A and o- the velocities associated with the corresponding modes, tOo is a measure of the restoring force for the polar vibrations, the parameter c gives the strength of the non-linear coupling between the two types of displacements, while the last term (with strength d ~) is a quartic anharmonic term inserted to keep open the possibility of the softening of polar vibrations (without the disruption of the system). Introducing the relative displacement or compressional deformation ~b-=-(OA/cTx), the system equations b e c o m e 2
~p
3 P ,2 3 2 dEp3 - ~ r - ~ -~-Z + tooP - c~bP + =0,
(2a)
o~2~b 2 02~b c 02 p2 --ff/r - or ~ +~~ = O.
(2b)
It is expected that the elastic field changes at a m u c h slower pace than the polar mode, and to keep track of this fact it is convenient to go over to the phase representation, namely P = N + ~ * , and to omit all antiresonant terms to obtain the coarse-grained time-scale equations of motion 02~
_
h 2 a2~ --~-~-x+ tooz~ +3d21.~I2~ - c ¢ ~
=0
(3a)
a2~b 2 a2~b ~2 - ~ r - or ~ + c ~-~ I~[ 2 = 0,
(3b)
and the corresponding effective Lagrangian density
~LP~e= ~ * ~ - to2~*~ - 2 d 2 ~ . 2 ~ 2
Ox
Ox
-2-\Ox/
(OA)
- c -~-
~*~.
(4)
Invariance under space and time displacements as well as under gauge transformations ( ~ ~ e iA~, ~ , ~ e iA~, for arbitrary real A) gives rise to associated conserved quantities which are the usual m o m e n t u m , energy and the 'polaron' number (N), where m = - if
dx(~*
- ~*~),
(5)
708
D. BHAUMIK, B. DUTTA-ROY AND A. LAHIRI
the usefulness of which will appear later. Since we are searching for solutions describing localised excitations m o v i n g with s o m e velocity, say v, it behaves us to introduce the variable ~ = x - vt. It is e x p e c t e d that the slow field ~b is a function of ~: alone, w h e r e a s ~ should have a slow modulation, PR(O on its intrinsic rapid variation, enabling us to claim d~ = d~(~), ~ ( x , t) = Pk(~) exp [i(kx - tokt)],
(6)
which, when incorporated in (3), gives (A 2 _ v2) -dZpk - ~ + (toZk -- tOO2 -- A2k 2 - 3 d 2 + Cch)pk = O, d2~b
(o'2-
c
d2
0,
(7a)
(7b)
supplemented by the dispersion relation (7c)
tok = A 2k/v.
The solution of (7b), n a m e l y ~b = ~bo + [c/(o. 2 - v2)]p~,
(8)
where ~bo represents an externally i m p o s e d u n i f o r m strain, w h e n substituted in (7a) leads to the e q u a t i o n d~--~
A2_v 2
Pk+(AZ--v2)(o.Z_v2
)
v2±~-d~--o. Jpk=O,
(9) which possesses a localised solution Pk = a sech bE,
(10)
where ( - b 2) is the coefficient of the linear term and (2bZ/a z) that of the cubic term in (9). Reality of a and b (with)t > or as is e x p e c t e d to be the case) requires that the velocity lies in the range o'2> v2> (o . 2 - c2[3d 2) (omitting from the discussion the physically uninteresting case v > )t). In the situation where w e a k coupling (c small) and strong non-linearity (large d) holds, the velocity of the solitary w a v e lies in a n a r r o w b a n d near the sound velocity, with the amplitude diverging at the lower edge
SOLITARY WAVES AND MACROMOLECULAR SYSTEMS
709
and vanishing at the velocity of sound. A less singular, more interesting and useful eventuality is realised for molecules possessing strongly coupled modes and weak non-linearity (with c 2 / 3 d 2 - t r 2 > O ) when a localised excitation could travel with any velocity between 0 and tr, depending on the details of its genesis, and a wide band pass system is obtained. Furthermore, localisability (reality of b) necessitates the positivity of (oJ02- C~bo)/()t 2 - v 2) - A2k2/v2, which may be guaranteed if o~02> C~bo and, moreover, the wave-number is not too large. On the other hand, the time-scale argument rules out wave-numbers that are too small by virtue of the dispersion relation, (7c). Hence the existence of optimal conditions for the generation of solitary waves. For an arbitrary initiating disturbance characterised by an energy, m o m e n t u m and extent of localisation, one would expect, in general, the production of travelling c noidal wave-trains corresponding to a solution of (9) in terms of the elliptic function cn. However, for the velocity parameter v, lying in the regime delineated above, these wave-trains undergo modulations and tend towards the solitary wave solution. The system may, however, search out a more involved solution (Mima et al., 1977) obtained by a Fourier decomposition of the polarisation field in place of (6), with a sum over wave-numbers, posing a problem which is still tractable provided one employs the random phase approximation, and then the general solutions are available as associated Legendre polynomials pT(tanh b~:), the special case S = 1, m = 1 being the solution explicitly depicted in (10). Furthermore, an important aspect of the systems equation (3) is that there exists a set of exact solutions, namely ~ = 0 and ~b satisfying the standard undriven linear wave equation giving pure elastic wave-trains, which, it may be noted, contribute nothing to the 'polaron n u m b e r ' (i.e. N = 0) as defined by (5), although they add finite amounts to the momentum and energy integrals. The value of N, on the contrary, for solitary waves, given by (10), is non-vanishing and given by toa2/b. The conserved quantities (energy, m o m e n t u m and 'polaron number') gives rise to some restrictions (Thornhill and Ter Haar, 1978) on the interactions of elementary excitations, yielding i n t e r alia some relevant selection rules: (a) isolated solitons (used loosely here in place of solitary waves) cannot decay to pure elastic wave-trains (because of N conservation or gauge invariance); (b) an isolated soliton cannot break up into more than one soliton accompanied by elastic wave-trains (because for given values of N and the m o m e n t u m , the energy corresponding to a single solitary wave is a minimum); (c) a soliton together with an elastic wave-train can always make a transition to another solitary wave of different velocity with an associated emission of sound waves. We should emphasise, however, that these selection rules only tell us what can happen, not
710
D. BHAUMIK, B. DUTTA-ROY AND A. LAHIRI
what will happen, and it is in this sense that the relative stability of isolated localised excitations [equation (10)] should be discerned. A totally different situation occurs, however, when the energy residing in the polarisation mode is large enough to cause a sufficient attendant elastic deformation ~b0 to soften (Wo2 < c4,o) the polarisation mode. This could also be actuated by an adequate, externally imposed strain. In the stationary state description this would correspond to the onset of 'ferroelectric' metastability. For the present purpose this would m e a n that the variations of the polarisation field occurs about a level Po = ( Cqbo -
01)
~ ) l d 2,
whereby the softening of the m o d e counteracted by the quartic selfinteraction (preventing disruption of the system) has led to a displacement in the polarisation ('vacuum' in the language of field theory). It is appropriate in this event, then, to introduce (12)
~b = ~bo+ ~o,
P=Po+p;
together with the phase representation, p = ~ + ~ * , and to implement the time scale argument as before, to arrive at the equations of motion a2~ --}k 2 a2~ +2d2po2~ + 3d21~12~ - c ~ o ~ = O,
~-
(13a)
-~x
a2(~
2 32~ t~
- tr ~
E~2
+ c ~-~ I~[ 2 = O,
(13b)
along with an equation of constraint (13c)
~o = 6d21~12/c.
The corresponding 'effective' Lagrangian density is
~e=~,~_#a~*a~, ax
aS+T
A2 ( d ~ 4 °2
or -~3 d2~,,2~r~2 --~~2..{_
+ ' -
)
~ 4,~ - 2 d 2 e ~ * ~
Po2 - o'2~bo q~ + c~p~*~,
'
(14)
where the fields are appropriately constrained. Proceeding as previously to write ~ = p k ( ~ ) e x p [i(kx-tokt)] and ~ = ~(~:), we obtain the dis-
SOLITARY WAVES AND MACROMOLECULAR SYSTEMS
persion relation tok =
k,~2/1J
711
and the solution to (13b), = c p ~ / ( o "2 - v2)
(15)
which, together with the constraint equation (13c), fixes the value of v to be (16)
v 2 = or2 - c 2 / 6 d 2.
Finally, (13a) becomes dEpk + (A~vk2 d~2
2dEp2~
3d 2
---~2-~_v )pk d - ~
Pk = O,
(17)
which admits localised solutions of the form Pk = a sech b~, provided ( 2 d 2 p 2 [ A 2 - 1/2)> (,~2k2/1.'2) and the coupling constant is weak enough and the non-linearity sufficient to guarantee O ' 2 ~ c2/6d 2, needed for the reality of the velocitY. Further detailed analyses, including relevant selection rules, may be made in a manner analogous to the preceding case. We may infer that when the macromolecule has been driven to the 'ferro-electric' state (with giant static electric dipole moment) against this background polarisation, localised travelling excitations are still possible (provided c 2 / 6 d 2 < irE), although here the velocity is fixed in terms of the effective parameters of the molecules. Conclusion. In conclusion, this paper is dedicated to the need felt in bioenergetics to develop models for spatial energy transfer along protein molecules. Solitary waves convey with efficiency the energy of excitation between different portions of a macromolecule. Furthermore, since these localised excitations are in the form of polarisation 'lumps', they possess the potentiality of trapping ions and thereby could provide a mechanism for ion transport. The excitation may be triggered by chemical energy released in the hydrolysis of adenosine triphosphate, which is the universal energy 'currency' of biophysics, or could be induced by the notion of an external electric field due to the approach of a charged ion. It has been shown that a rich structure of solitary waves emerges, including the possibility of envelope 'solitons', which could be made use of by the macromolecule depending on its state as well as the characteristics of the initiation process. The widest range of conditions for the occurrence of travelling localised excitations hold when the molecule is in its normal state and is one which possesses weakly non-linear polar modes strongly coupled to the elastic vibrations. On the other hand, when the molecule
712
D. BHAUMIK, B. DUTTA-ROY AND A_ LAH1RI
goes o v e r into the ' f e r r o - e l e c t r i c ' s t a t e as a r e s u l t o f e x t e r n a l c o n s t r a i n t s , the solitary w a v e e n d o w e d w i t h a definite v e l o c i t y ( g i v e n in t e r m s o f the p a r a m e t e r s o f the m o l e c u l e ) is e x c i t e d w h e n t h e p o l a r m o d e has strong n o n - l i n e a r i t y a n d is w e a k l y c o u p l e d to the a c o u s t i c d e g r e e s o f f r e e d o m . This v a r i a b i l i t y in the ability to s u p p o r t s o l i t a r y w a v e s s e e m s to be a p a r t i c u l a r l y a t t r a c t i v e f e a t u r e , s i n c e the e f f e c t i v e p a r a m e t e r s p e r taining to the p o l a r a n d a c o u s t i c m o d e s o f m a c r o m o l e c u l e s or t h e i r aggregates d e p e n d , n o d o u b t , o n t h e s t a t e o f t h e i r e n v i r o n m e n t ( w i t h which t h e y are in a s s o c i a t i o n ) as, f o r e x a m p l e , the c a s e o f p r o t e i n molecules in the b i l a y e r lipid m e m b r a n e . H e n c e , the p o s s i b i l i t y o f t h e same m o l e c u l e p e r f o r m i n g d i f f e r e n t f u n c t i o n s u n d e r d i f f e r e n t c o n d i t i o n s (L~iuger, 1979) is an o u t c o m e o f o u r m o d e l w h i c h m a y b e o f c o n s i d e r a b l e use to biology. The a u t h o r s e x p r e s s t h e i r d e e p s e n s e o f g r a t i t u d e to B r a h m a n a n d a D a s g u p t a and T a r a s h a n k a r N a g f o r h e l p f u l d i s c u s s i o n s . LITERATURE Berteaud, A. J., M. Dardalhon, N. Rebeyrotte and M. D. Averbeck. 1975. "Action d'un Rayonnement Electromagnetique a Longueur d'Onde Millimetrique sur la Croissance Bacterienne". C. r. hebd. S~anc. Acad. Sci, Paris 281,843-847. Bhaumik, D., K. Bhaumik and B. Dutta-Roy. 1976a. "On the Possibility of 'Bose Condensation' in the Excitation of Coherent Modes in Biological System". Phys. Lett. 56A, 145-148. , - - and . 1976b. "A Microscopic Approach to the Fr6hlich Model of Bose Condensation of Photons in Biological Systems". Phys. Lett. 59A, 77-80. , - - , - - and M. H. Engineer. 1977. "Polar Modes with Elastic Restoring Forces, 'Bose Condensation' and the Possibility of Metastable Ferroelectric State". Phys. Lett. 62A, 197-200. - - , A. K. Roy and B. Dutta-Roy. 1978a. "On a Model and the Kinetics of Photo-enhanced Enzyme Reactions". Bull. math. Biol. 40, 719-726. , B. Dutta-Roy and A. Lahiri. 1978b. "The Van-der-Waals Interaction Between Loose Structures". Phys. Lett. 68A, 131-134. and .1978c. "Effect of the Intervening Medium and of Feeding of Energy into Polar Modes on Macro-molecular Interactions". Phys. Lett. 69A, 68-72. Davydov, A. S. 1977. "Solitons and Energy Transfer along Protein Molecules". J. theor. Biol. 66,379-387. Deryatkov, N. D. et al. 1974. "Scientific Session of the Division of General Physics and Astronomy, USSR Academy of Sciences". Soviet Phys. lisp. 16, 568. Frfhlich, H. 1968a. "Bose Condensation of Strongly Excited Longitudinal Electric Modes". Phys. Lett. 26A, 402--403. 1968b. "Long Range Coherence and Energy Storage in Biological Systems". Int. J. quant. Chem. 2,641. 1968c. "Storage of Light Energy and Photosynthesis". Nature, Lond. 219, 743-744. ,
,
-
-
SOLITARY WAVES AND MACROMOLECULAR SYSTEMS
713
1969. Theoretical Physics and Biology, Ed. Marois, p. 13. Amsterdam: North-Holland. 1970. "Long Range Coherence and the Action of Enzymes." Nature, Lond. 228, 1093. 1972. "Selective Long-range Dispersion Forces between Large Systems". Phys. Lett. 39A, 153-154. 1973. "Collective Behaviour of Non-linearly Coupled Oscillating Fields with Applications to Biological Systems". Collect. Phenomena 1,101-109. 1975a. "Evidence for Bose Condensation Like Excitation of Coherent Modes in Biological Systems". Phys. Lett. 51A, 21-22. -1975b. "The Extraordinary Dielectric Properties of Biological Materials and the Action of Enzyme". Proc. natn Acad. Sci. U.S.A. 72, 4211-4215. Grundler, W., F. Keilmann and H. Frfhlich. 1977. "Resonant Growth Rate Response of Yeast Cells Irradiated by Weak Microwaves". Phys. Lett. 62A, 463-466. L~iuger, P. 1979. "A Channel Mechanism for Electrogenic Pumps". Biochim. Biophys. Acta 552, 143-161. Mima, K., K. Kato and K. Nishikawa. 1977. "Self-consistent Stationary Density Cavities with Many Bounded Langmuir Waves". J. phys. Soc. Japan 42, 290-296. Thornhill, S. G. and D. Ter Haar. 1978. "Langmuir Turbulence and Modulational Instability". Phys. Rep. 43. Webb, S. J. and A. D. Booth. 1969. "Absorption of Microwaves by Microorganisms". Nature, Lond. 222, 1199-1200. and M. E. Stoneham. 1977. "Resonances between 101C10'2 Hz in Active Bacterial Cells as Seen by Laser Raman Spectroscopy". Phys. Lett. 60A, 267-269. , - - and H. Fr6hlich. 1977. "Evidence for Non-thermal Excitation of Energy Levels in Active Biological Systems". Phys. Lett. 63A, 407-408. Wu, T. M. and S. Austin. 1977. "Bose Condensation in Biosystems". Phys. Lett. 64A, 151-152. - - and .1978. "Cooperative Behaviour in Biological Systems". Phys. Lett. 65A, 74-76. RECEIVED 5-1-81
0092-8240/82/050705-09503.0010 Pergamon Press Ltd. © 1982 Society for Mathematical Biology
SOLITARY WAVES AND MACROMOLECULAR SYSTEMS DEBAJYOTI BHAUMIK Bose Institute, 93/1 A c h a r y a Prafulla Chandra Road, Calcutta 700009, India BINAYAK DUTTA-ROY
Saha Institute of Nuclear Physics, 92 Acharya Prafulla Chandra Road, Calcutta 700009, India AvIJIT LAHIRI Vidyasagar (Evening) College,
Calcutta 700006, India
Macromolecules and their aggregates (such as protein bundles in biomembranes) possess polar modes which, when excited, tend to deform the system and call into play elastic restoring forces. A model of such systems, characterised typically by electric polarisation modes stabilised on the one hand by quartic self-interactions and on the other through coupling to the elastic deformations, admits the possibility of localised excitations (solitary waves) propagating with subsonic velocities, possessing the features of relative stability and efficient transport characteristics (associated with the collective nature of the phenomena), and at the same time provides a mechanism of control and variability which could be of considerable interest in biology.
Introduction. The propagation of localised excitations along protein molecules is often considered to play an important role in transport and transduction processes occurring in biology. The antagonists of this view-point, however, draw attention to the smallness of the life-times of such excitations, which would indeed be the case if these were associated with local chemical groups; on the other hand, it is our contention that the existence of collective modes in macromolecules (or their aggregates), with concommitant non-linearities makes possible relatively stable localised excitations, providing interesting mechanisms for transport. Indeed, the potentiality of solitary waves in adopting the role of such carriers has been emphasised by Davydov (1977). Nevertheless, the full richness of the underlying structure and the scope for control and influence by external factors, features of paramount importance in biology, has been left unexposed by that author. The object of the 705
706
D. BHAUMIK, B. DUTTA-ROY AND A. LAHIRI
present work is to uncover some of these aspects within the context of a model which has yielded some insight into the dynamics of macromolecules. In a series of papers, Fr6hlich (1968a, 1968b, 1968c, 1969, 1970, 1972, 1973, 1975a, 1975b) has conjectured that macromolecules often possess polar modes which, when excited, tend to deform the system in various ways, calling into play and counteracted by elastic forces which determine the shape of the unexcited system. The present authors (Bhaumik et al., 1976a, 1977, 1978a, 1978b) and others (Wu and Austin, 1977, 1978) have applied these ideas to discuss the possibility of a metastable 'ferro-electric' state with large dipole moment, consequent modification of the Van der Waals interaction between such molecules and implications on the mechanism of enzyme reactions. An appreciable body of experimental studies seem to suggest existence of collective polar modes of macromolecular systems and their role in biological phenomena. The extraordinary dielectric properties of biological materials (Fr6hlich, 1975b) emphasised by Fr6hlich, in particular the low lying vibrations in 1011 Hz frequency region (presumably with giant dipole moments), profound non-thermal effects (including marked enhancement of biological activity) by millimeter irradiation on biological samples (Webb and Booth, 1969; Deryatkov et al., 1974; Berteaud et al., 1975; Grundler et al., 1977) and results of laser raman spectroscopy (Webb and Stoneham, 1977) strongly suggest the existence and functional importance of collective polar modes of macromolecules. In our previous investigations attention was confined to the stationary configurations of the molecules, while this study is devoted to dynamical behaviour, particularly to the movement through the macromolecule of localised excitations.
The Model. The model (Bhaumik et al., 1977) is defined in terms of polarisation and acoustic fields P(x, t) and A(x, t), which are functions of time t and position x in the macromolecule (for simplicity, the coordinate along the chain has been taken as the only relevant dimension, and as vibrations in large molecules are frequently isotropic, the vectorial character of the fields has been suppressed). No doubt this description mutilates the detail of the chemical and stereo-structure of the molecule, but it provides a description which brings into focus the role of collective modes in the dynamical behaviour of the system, which is thus taken to be governed by the Lagrangian density 21[~2 + =
A 2 (c~P~ 2 _ o "2 (~A~ 2 1 2 2 - T
T
\ ax /
-
O oI. -
c (c~A~2p2
1 d2p4" --g
(1)
SOLITARY WAVES AND MACROMOLECULAR SYSTEMS
707
The first two terms represent the kinetic energy densities, A and o- the velocities associated with the corresponding modes, tOo is a measure of the restoring force for the polar vibrations, the parameter c gives the strength of the non-linear coupling between the two types of displacements, while the last term (with strength d ~) is a quartic anharmonic term inserted to keep open the possibility of the softening of polar vibrations (without the disruption of the system). Introducing the relative displacement or compressional deformation ~b-=-(OA/cTx), the system equations b e c o m e 2
~p
3 P ,2 3 2 dEp3 - ~ r - ~ -~-Z + tooP - c~bP + =0,
(2a)
o~2~b 2 02~b c 02 p2 --ff/r - or ~ +~~ = O.
(2b)
It is expected that the elastic field changes at a m u c h slower pace than the polar mode, and to keep track of this fact it is convenient to go over to the phase representation, namely P = N + ~ * , and to omit all antiresonant terms to obtain the coarse-grained time-scale equations of motion 02~
_
h 2 a2~ --~-~-x+ tooz~ +3d21.~I2~ - c ¢ ~
=0
(3a)
a2~b 2 a2~b ~2 - ~ r - or ~ + c ~-~ I~[ 2 = 0,
(3b)
and the corresponding effective Lagrangian density
~LP~e= ~ * ~ - to2~*~ - 2 d 2 ~ . 2 ~ 2
Ox
Ox
-2-\Ox/
(OA)
- c -~-
~*~.
(4)
Invariance under space and time displacements as well as under gauge transformations ( ~ ~ e iA~, ~ , ~ e iA~, for arbitrary real A) gives rise to associated conserved quantities which are the usual m o m e n t u m , energy and the 'polaron' number (N), where m = - if
dx(~*
- ~*~),
(5)
708
D. BHAUMIK, B. DUTTA-ROY AND A. LAHIRI
the usefulness of which will appear later. Since we are searching for solutions describing localised excitations m o v i n g with s o m e velocity, say v, it behaves us to introduce the variable ~ = x - vt. It is e x p e c t e d that the slow field ~b is a function of ~: alone, w h e r e a s ~ should have a slow modulation, PR(O on its intrinsic rapid variation, enabling us to claim d~ = d~(~), ~ ( x , t) = Pk(~) exp [i(kx - tokt)],
(6)
which, when incorporated in (3), gives (A 2 _ v2) -dZpk - ~ + (toZk -- tOO2 -- A2k 2 - 3 d 2 + Cch)pk = O, d2~b
(o'2-
c
d2
0,
(7a)
(7b)
supplemented by the dispersion relation (7c)
tok = A 2k/v.
The solution of (7b), n a m e l y ~b = ~bo + [c/(o. 2 - v2)]p~,
(8)
where ~bo represents an externally i m p o s e d u n i f o r m strain, w h e n substituted in (7a) leads to the e q u a t i o n d~--~
A2_v 2
Pk+(AZ--v2)(o.Z_v2
)
v2±~-d~--o. Jpk=O,
(9) which possesses a localised solution Pk = a sech bE,
(10)
where ( - b 2) is the coefficient of the linear term and (2bZ/a z) that of the cubic term in (9). Reality of a and b (with)t > or as is e x p e c t e d to be the case) requires that the velocity lies in the range o'2> v2> (o . 2 - c2[3d 2) (omitting from the discussion the physically uninteresting case v > )t). In the situation where w e a k coupling (c small) and strong non-linearity (large d) holds, the velocity of the solitary w a v e lies in a n a r r o w b a n d near the sound velocity, with the amplitude diverging at the lower edge
SOLITARY WAVES AND MACROMOLECULAR SYSTEMS
709
and vanishing at the velocity of sound. A less singular, more interesting and useful eventuality is realised for molecules possessing strongly coupled modes and weak non-linearity (with c 2 / 3 d 2 - t r 2 > O ) when a localised excitation could travel with any velocity between 0 and tr, depending on the details of its genesis, and a wide band pass system is obtained. Furthermore, localisability (reality of b) necessitates the positivity of (oJ02- C~bo)/()t 2 - v 2) - A2k2/v2, which may be guaranteed if o~02> C~bo and, moreover, the wave-number is not too large. On the other hand, the time-scale argument rules out wave-numbers that are too small by virtue of the dispersion relation, (7c). Hence the existence of optimal conditions for the generation of solitary waves. For an arbitrary initiating disturbance characterised by an energy, m o m e n t u m and extent of localisation, one would expect, in general, the production of travelling c noidal wave-trains corresponding to a solution of (9) in terms of the elliptic function cn. However, for the velocity parameter v, lying in the regime delineated above, these wave-trains undergo modulations and tend towards the solitary wave solution. The system may, however, search out a more involved solution (Mima et al., 1977) obtained by a Fourier decomposition of the polarisation field in place of (6), with a sum over wave-numbers, posing a problem which is still tractable provided one employs the random phase approximation, and then the general solutions are available as associated Legendre polynomials pT(tanh b~:), the special case S = 1, m = 1 being the solution explicitly depicted in (10). Furthermore, an important aspect of the systems equation (3) is that there exists a set of exact solutions, namely ~ = 0 and ~b satisfying the standard undriven linear wave equation giving pure elastic wave-trains, which, it may be noted, contribute nothing to the 'polaron n u m b e r ' (i.e. N = 0) as defined by (5), although they add finite amounts to the momentum and energy integrals. The value of N, on the contrary, for solitary waves, given by (10), is non-vanishing and given by toa2/b. The conserved quantities (energy, m o m e n t u m and 'polaron number') gives rise to some restrictions (Thornhill and Ter Haar, 1978) on the interactions of elementary excitations, yielding i n t e r alia some relevant selection rules: (a) isolated solitons (used loosely here in place of solitary waves) cannot decay to pure elastic wave-trains (because of N conservation or gauge invariance); (b) an isolated soliton cannot break up into more than one soliton accompanied by elastic wave-trains (because for given values of N and the m o m e n t u m , the energy corresponding to a single solitary wave is a minimum); (c) a soliton together with an elastic wave-train can always make a transition to another solitary wave of different velocity with an associated emission of sound waves. We should emphasise, however, that these selection rules only tell us what can happen, not
710
D. BHAUMIK, B. DUTTA-ROY AND A. LAHIRI
what will happen, and it is in this sense that the relative stability of isolated localised excitations [equation (10)] should be discerned. A totally different situation occurs, however, when the energy residing in the polarisation mode is large enough to cause a sufficient attendant elastic deformation ~b0 to soften (Wo2 < c4,o) the polarisation mode. This could also be actuated by an adequate, externally imposed strain. In the stationary state description this would correspond to the onset of 'ferroelectric' metastability. For the present purpose this would m e a n that the variations of the polarisation field occurs about a level Po = ( Cqbo -
01)
~ ) l d 2,
whereby the softening of the m o d e counteracted by the quartic selfinteraction (preventing disruption of the system) has led to a displacement in the polarisation ('vacuum' in the language of field theory). It is appropriate in this event, then, to introduce (12)
~b = ~bo+ ~o,
P=Po+p;
together with the phase representation, p = ~ + ~ * , and to implement the time scale argument as before, to arrive at the equations of motion a2~ --}k 2 a2~ +2d2po2~ + 3d21~12~ - c ~ o ~ = O,
~-
(13a)
-~x
a2(~
2 32~ t~
- tr ~
E~2
+ c ~-~ I~[ 2 = O,
(13b)
along with an equation of constraint (13c)
~o = 6d21~12/c.
The corresponding 'effective' Lagrangian density is
~e=~,~_#a~*a~, ax
aS+T
A2 ( d ~ 4 °2
or -~3 d2~,,2~r~2 --~~2..{_
+ ' -
)
~ 4,~ - 2 d 2 e ~ * ~
Po2 - o'2~bo q~ + c~p~*~,
'
(14)
where the fields are appropriately constrained. Proceeding as previously to write ~ = p k ( ~ ) e x p [i(kx-tokt)] and ~ = ~(~:), we obtain the dis-
SOLITARY WAVES AND MACROMOLECULAR SYSTEMS
persion relation tok =
k,~2/1J
711
and the solution to (13b), = c p ~ / ( o "2 - v2)
(15)
which, together with the constraint equation (13c), fixes the value of v to be (16)
v 2 = or2 - c 2 / 6 d 2.
Finally, (13a) becomes dEpk + (A~vk2 d~2
2dEp2~
3d 2
---~2-~_v )pk d - ~
Pk = O,
(17)
which admits localised solutions of the form Pk = a sech b~, provided ( 2 d 2 p 2 [ A 2 - 1/2)> (,~2k2/1.'2) and the coupling constant is weak enough and the non-linearity sufficient to guarantee O ' 2 ~ c2/6d 2, needed for the reality of the velocitY. Further detailed analyses, including relevant selection rules, may be made in a manner analogous to the preceding case. We may infer that when the macromolecule has been driven to the 'ferro-electric' state (with giant static electric dipole moment) against this background polarisation, localised travelling excitations are still possible (provided c 2 / 6 d 2 < irE), although here the velocity is fixed in terms of the effective parameters of the molecules. Conclusion. In conclusion, this paper is dedicated to the need felt in bioenergetics to develop models for spatial energy transfer along protein molecules. Solitary waves convey with efficiency the energy of excitation between different portions of a macromolecule. Furthermore, since these localised excitations are in the form of polarisation 'lumps', they possess the potentiality of trapping ions and thereby could provide a mechanism for ion transport. The excitation may be triggered by chemical energy released in the hydrolysis of adenosine triphosphate, which is the universal energy 'currency' of biophysics, or could be induced by the notion of an external electric field due to the approach of a charged ion. It has been shown that a rich structure of solitary waves emerges, including the possibility of envelope 'solitons', which could be made use of by the macromolecule depending on its state as well as the characteristics of the initiation process. The widest range of conditions for the occurrence of travelling localised excitations hold when the molecule is in its normal state and is one which possesses weakly non-linear polar modes strongly coupled to the elastic vibrations. On the other hand, when the molecule
712
D. BHAUMIK, B. DUTTA-ROY AND A_ LAH1RI
goes o v e r into the ' f e r r o - e l e c t r i c ' s t a t e as a r e s u l t o f e x t e r n a l c o n s t r a i n t s , the solitary w a v e e n d o w e d w i t h a definite v e l o c i t y ( g i v e n in t e r m s o f the p a r a m e t e r s o f the m o l e c u l e ) is e x c i t e d w h e n t h e p o l a r m o d e has strong n o n - l i n e a r i t y a n d is w e a k l y c o u p l e d to the a c o u s t i c d e g r e e s o f f r e e d o m . This v a r i a b i l i t y in the ability to s u p p o r t s o l i t a r y w a v e s s e e m s to be a p a r t i c u l a r l y a t t r a c t i v e f e a t u r e , s i n c e the e f f e c t i v e p a r a m e t e r s p e r taining to the p o l a r a n d a c o u s t i c m o d e s o f m a c r o m o l e c u l e s or t h e i r aggregates d e p e n d , n o d o u b t , o n t h e s t a t e o f t h e i r e n v i r o n m e n t ( w i t h which t h e y are in a s s o c i a t i o n ) as, f o r e x a m p l e , the c a s e o f p r o t e i n molecules in the b i l a y e r lipid m e m b r a n e . H e n c e , the p o s s i b i l i t y o f t h e same m o l e c u l e p e r f o r m i n g d i f f e r e n t f u n c t i o n s u n d e r d i f f e r e n t c o n d i t i o n s (L~iuger, 1979) is an o u t c o m e o f o u r m o d e l w h i c h m a y b e o f c o n s i d e r a b l e use to biology. The a u t h o r s e x p r e s s t h e i r d e e p s e n s e o f g r a t i t u d e to B r a h m a n a n d a D a s g u p t a and T a r a s h a n k a r N a g f o r h e l p f u l d i s c u s s i o n s . LITERATURE Berteaud, A. J., M. Dardalhon, N. Rebeyrotte and M. D. Averbeck. 1975. "Action d'un Rayonnement Electromagnetique a Longueur d'Onde Millimetrique sur la Croissance Bacterienne". C. r. hebd. S~anc. Acad. Sci, Paris 281,843-847. Bhaumik, D., K. Bhaumik and B. Dutta-Roy. 1976a. "On the Possibility of 'Bose Condensation' in the Excitation of Coherent Modes in Biological System". Phys. Lett. 56A, 145-148. , - - and . 1976b. "A Microscopic Approach to the Fr6hlich Model of Bose Condensation of Photons in Biological Systems". Phys. Lett. 59A, 77-80. , - - , - - and M. H. Engineer. 1977. "Polar Modes with Elastic Restoring Forces, 'Bose Condensation' and the Possibility of Metastable Ferroelectric State". Phys. Lett. 62A, 197-200. - - , A. K. Roy and B. Dutta-Roy. 1978a. "On a Model and the Kinetics of Photo-enhanced Enzyme Reactions". Bull. math. Biol. 40, 719-726. , B. Dutta-Roy and A. Lahiri. 1978b. "The Van-der-Waals Interaction Between Loose Structures". Phys. Lett. 68A, 131-134. and .1978c. "Effect of the Intervening Medium and of Feeding of Energy into Polar Modes on Macro-molecular Interactions". Phys. Lett. 69A, 68-72. Davydov, A. S. 1977. "Solitons and Energy Transfer along Protein Molecules". J. theor. Biol. 66,379-387. Deryatkov, N. D. et al. 1974. "Scientific Session of the Division of General Physics and Astronomy, USSR Academy of Sciences". Soviet Phys. lisp. 16, 568. Frfhlich, H. 1968a. "Bose Condensation of Strongly Excited Longitudinal Electric Modes". Phys. Lett. 26A, 402--403. 1968b. "Long Range Coherence and Energy Storage in Biological Systems". Int. J. quant. Chem. 2,641. 1968c. "Storage of Light Energy and Photosynthesis". Nature, Lond. 219, 743-744. ,
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