Coferent dynamics in biomolecular systems

Journal ofMdecular

Liquids,4i(i08!3)~166-180 EY#vieTscieI~PublishereB.V, Ammkdmm-Prir~tadinTheNetherlands

COI-FERENT DYNAMICS U’DLIAM

IN BXOMOLECULAR

166

SYSTEMS

RHODES

De artnxnt of Chcmisay and Instituteof Molecular Biophysi&, Florida State University. T a# ahassc~Fkuida3230S

ABsrRAcr S&c of the chflkultics in devel~ng fi.~ndamcnud concepts and principles for the dynamical aspects of molccula~levcl b~ol?~gyare discussed_ Use is made of similarities between phyrncalprocesses involved in biomolecular mechanisms and thosa encountered commonrobottlafcpointcdouSand in molecular spcctroscopy~ Concepts and p it is shown how the approach we used forTrti=vcloplng reduced subsystem equations fm Im~tfeahlrcs ic pxxcscs is applicable also for biomolcc+ar_ pv. specm include“p (1 a fundamental cbchotomy leading to the m and Qlimits of subsystem dynamics and (2) the distinction between equilibrium diffusion and uilibriumflow in hasc (Hilbut) space. The concept of Macromola%ll Activation as a p” undamcntal enuty in biomolccular mechanisms ix pXScntcdn$K%Qcles Conjuxurcs am made regardingboth the.thcxmodynamicand uantummcchamcal mtture of MAC cycles. The a-helix soliton model of Davydov. CarelI. end Scott is rtformulatcd without using the assumptions that produce tha nonlincfu Schroedingcr equation: Calculations show thatthe soliton dots not exist in the a-helix; rather,theexcitor migmtts mora in accxxd with the dissipativelimit. INTROIkTION One of the most subtle and difficult ptiblcms facing the physical and biological sciences concerns thaquestion: What are the underlyingconcepts and principleswhereby complex assemblies of biomoleculcs opuate to carry out biological fun&&s?

Most of

US

agree thatphysics and chemistryprovidua bssis for understandingmolecular-level biologyBut thequestion is whether existing concepts and principlesof physical th&ny as they apply to chemistry and solid-state physics. for example. arc sufficient - or do they havu to be recombined and rcshap& &to somethingrecognizably new aid different?

The background

for this question is clear when we observe the eagerness with which every new idea of the physical and mathematical sciences is transferredto biology; for cxampl~ - elec&~ cclnductionbands. excitons, solitons, strange attractors.and fiactals, to name a few:.. Yet none of thy

%a,

thus far. offers an underlyingprincipleor paradigmfor biology;

Oli37-7322/89/$&50

8 19fIQElsevierSciencePubliaherpB-V_

166

Fromanothtrptrspbctive.b~olo~calsyste~maybtvIcwcdascxtcnsionsof of systems t&ted by &densed &p&h

matter physics.

thtkinds

As such, they arc cond&tscd systems which

to h&c a high degree of structural heterogeneity and. accordingly. thay bring to bear

all of the problems and difficulties encountered in the theory of large systems having many degrees of freedom_ The latter includes the formuladon of appropriate subsystem equations of motion and the developmetit of mcthod6logiw

for quantitativ&ly applying such equations

to reasonable models for real systems. Over the years, numerous physical scientists have devoted their attention in part or in whole, to the fundamen tal aspects of biological systcmS_ Giorgio Car&

[ I] is one of thcsa

rcscarchtiwhohasmadccxtcnsiveapplicationsofphysicalconcc$sandmodels

tobiology!

It is appropriate that a special issue of this journal be dedicated in his honor. The relation of our work to his on soliton dynamics in polypeptidcs will be discussed below. It is our view that the existing concepts and principles of physical theory provide the building blocks from which an understanding of biology can be constructed_ Thult is no intellectual demur!

that we go beyond

the fiarnework

of statistical thermodynamics,

qua&urn and classical mechanics, and field theory_ To be; sure many. rl3xarchershavcbccn self-limited in their applications of the than-y. r&ulting in misconceptions and too-severe restrictions on the nature of the system and, consequently, calling for a drastic departure .fivmcurfentthcory_ While

we do not believe that a departure from cumnt

thcoty is necessary, we do

believe that &tensions 3nour thinking about hoti theory applies to microtipic. level dynamics is required (the reshaping r&red understanding the basic functions of biology

to above)_

molecular

We are a long way from

bmch as, &If-assembly.

organization,

and

replication. The mechanisms whereby energy and information flow arecoordinated remain amystery. New couccpts of association arecalled for, relatingmicroscopicand macroscopic live1 dynamics of highly-organized

molecular assemblies. 77iat arc few, if tiy. principles

Ft known that ar& unique to the dynamic behavior of biomolecular systems. The purpose of this paper is to put forth our views on fundamental biomolccular

dynamic,

macromolecules,

Focus

is placed on the role of cyclic

asp&s

processes

of

in which

such as proteins, operate by going through clectronic-confmmational

cycles. WC refer to these as macromolazulc activation cycles (MAC aspects of such cycles are of key importance.

cycles)_ Tht~

A maj& point of the paper is to present

conjtiturcs concerning the them&dynamic and quantum mechanical nature of MACcycles. Our views on biomolccular dynamics have been influenced by our ongoing ~heorctical studies on ni~lccularexcitation and relaxation prxesses of thctyptencountercd in molecular spectroscopy_ These studies have led us to consider the fundamental equations of modon for a molecular subsystem whose dynamics are modulated by other subsystems and by a

The following

background which may act as a dissipative medium on these points amcaming

the conceptual

fundamental subsyottm cqt&io~~

thrca sections elaborate

interplay among mOlccular spectroscopy.

ofmotion. and biomolecular

dynamics (MAC

cyclc~).

BACKGROUND Biological thumodynamic

systems am driven

subsy&ms

which

arc both nonequillbrium

in a

sense and nonstationary In a quantum mechanical sense_ Overall. they arc

driven by their en vironmcnt which serves as both source and sink with rtsptct to energy and matter flow. Intunall~4 a biological system (ag.. a cell) is further composed of coupled subsystems which drive one another according to a hiuaxchkal extremely high degree dsuuctural

pamm.

This makes for an

heterogeneity whereby there may be only kc

macromolcculcsofagivcnldndinaparticul~scructurt,

or a few

Succcssfulapplicatio~ofquantum

must take ink atxount &is aaanp&& nature-

theory and statistical Mynamics

On the other hand, many of the systems and phenomena of molecular specaoscopy are quite similar nonequilibrium

to biological

subsystems,

systems in that they involve

whose

dynamics

arc modulated

driven, by

nonstationary,

their cnvironuxnt.

S~lthcorIwofmoltcularspectroscoWmustincludtthecompo9i~nanrrcofsysrtms withrcgal7I to the various kinds of moIccularstructurcs

and thtla&ation

fxld,

It is reasonable to expect that most of the contipts and physical features that arc opcmfivc in spectroscopic processes arc aso important in biological proasses.

Following

are some

ofthemoreimportantofthcse. (1)

m

is perhaps the m&t fundamental quantum mechanical concept

It refkrs

to the existence of a dcfinita phase relation in th& probability amplitudes for the values of obscrvablcs.

It is the basis of tbc Superposition Principle and of &zr&cn&

effi

in

quantum mechanics_ (2)

&mtummechanical

-refers

to thedepcndcnct of &probability

amplitude

of one observable on the values of another. It involves the concept of e_ Each coordinate (and each spin) variable of each particle in a system is a ~&IU Furthermore. each photon mode is a dcgrce of worn.

of freedom.

composite systems consist of .&any

degrees of freedom, usually of diffkrcnt types_ In general. the states of a real system involve a high degree of corrcl~tion akng degrees of frudoti A &en

FLirthc~,

its

degrees of freed&m arc combined to form m.

subsystem may be a molecule, a part or a molcculc (a &mm&ho&

group), or a collccdon of molecules. The &&

of freedom which co&&c

or a substitucnt a subsystem

a& sor&imcs

referred to as mu&.

For the states of real systems the v

bccorrelatcd.

It is such corxclation which leads to the meaningful conctpts of v

168

A g&d example of the physi.Cal cff&ts of the tim&cvolution of subsystem &rmlation is found in the excitation of mol&ulcs cm&ping

by stationary light which has a frcqucncy spectrum

mom than ant ahsorption frequency for transitions from a given molecule state

[2]_ With incmasing t. them is in creasing correlation between the axcited-molecule and the frequency of the absorlxd bccouic ir&casingly is lost

statisd~y

photon,

is that the excited statis

hidependent. behaving as though coherence between them

Each uccitcd state fluoresces

molecule-field

The physical &f&t

state

comzlation sets in.

Howcvur.

independently.

for small t, bcfom

thaexcitation forms a coherent superposition of excited

states which exhibit interference effects in the form of so-called quantum beats in emission

131. (3) m

of sf~ff=s. and dilution

arc concepts which. in addition to

correlation, arise from the existence of more than one dcgrcc

of freedom

The fact that the

Hilbcrt space for a Composite system is spanned by a climct product basis for all degrees of fxtxilommcans that each eigenvaluc for a given degree of fmcdom. or subsystum, has many cigcnvcctors m.

Interactions among the dcgnx%s of freedom v

remove the dtgenuacy,

can

thus leading to the cigcnvcctors for a given subsystem Cigenvaluc

being spread over a range of energy (&t&y

nf ,s

concept)_ Consequently. transition

strengths corrcspondilrg to the given subsystem cigenvaluc- become M

for transitions

to any one energy level. For transitions between two given subsystem states. dilution ef%cts can occur in either the initial or the final state, An cxamplt of the forma [24]

is the uncxpectcdly long Iifetimeo

(associated with excitation of statistically independent excited states as discussed

above) of fluorescence

of S&and

NG

and of the latter is the kinetic slowing of radiadonlczu

transitions due to spectral broadening [Z,!TJ. Note ‘that such dilution cffixts

arc intimately related to the correlation

cfficts

mentioned above, An example of final state degeneracy effixts is formed in coherent light scattering by an cnsemb!e of molecules_ For Rayleigh scattering the initial and final state of thi molecule system is the same. Each molccul~ contributes a probability amplitude to photon scattering. thus, the probability involves interferences among the m&cults. other hand, for Raman scattering. one moleculi

CIn the

is vibrationally excited in the final state.

The scattered photon does not care which molecule is excited (final state dcgenctacy)_ H&ever,

onIy that molecule which is excited can scatter the photon. so there arc no

interference ~crms; the scattering probability is simely the sum of the probabilities for the mOlCCUlCS. (pjam

c&elation

-

i-&d-

v

-

-

arc fiirthn

15-71. Often. we arc interested in the dynamics

consequences

of subsystem

ofa particular subsystem,

we call the Primary Subsystem (K’S)_ The remainder of the total system is Background (EG).

In the stochas&

(or dissipative) hrnit. the &relation

which

termed the function for tha

..

189. PS-BG

interaction & aI&ac

delta fimctiori in time [S]. This n%ans that the coherence terms

hetwecn PS state relax nponcntially

(w

relaxation) and. for fast decay; the result is

kiuetic transitions among PS mtcs (rather than coherent dynamics).

h&cular

relaxation

~KKUECS such as quantum heats and intramolecular vibrational xdistributioi~ (IVR) greatly affix%& by PS-BG (5)

Adiabetic

art

modulation.

is one of the most usecful concepts in molecular physics-

One

subsystem (slow) movcoinapottntial~bythuavuagamotionofanorhersubsystGm (f&t)_

The Born-Uppa&Cmer

model. wha-cby

potential given by the electron ic enagY conceptual framework

plus Internuclear Coulomb potential. tides

of structural chun&xy

electronic transitions are conamed

the vilmuions of a molecule move in a and molazular

vopy:

the

Asfara!z

one may say that the dynamics of electronic motion are

modulated by motion of the nuclei. This leads to the FranckCondon

Principle_

WC have deyelopcd au adiabatic modylation aplproach to the efi?cts of the BG or mediumon the dyuatniw of amolecult;

Another example

as a basis forunmg

dissipation 13.581.

of adiabatic modulation is the soliton model proposed by Davydov

[9] and modified by Can+, of proteins.

ix,

et aL. [lo]

Here. a vi&ational

involving thahydrogen-bonding

and by Scott [ 1 L] for energy migration in the *helix

e&ton

motion is modulated by a low-frcqucnj

proton &f the &-helix, This model AI

be discu&d

example of_-

bel~sufficeittosayherethatthisisan

mode f&th&

iti which the cxciton

and lattice vibration move in concerL Above anz three examples of adiabatic modulation in mol&lar which the physical consequence is radically diff&ent:

dynamics, in each of

Fmnck-Condon

&ecu-al structure.

dissipative relaxation, producing diphasing (coherence relaxation) effects and entropy. and concerted (soliton) motion of diff&ent kinds of degrees of &uiom

SUBSYSTEM EQUATIONS A sy&n concsponding

(modes).

OF MOTION

in a pure quantura state is represented by a single vector I u(r) z-. state (density) operator is Hf)

=

1 u(t) x

u(c) 1. On the otkr

system in a statistically mixed state is represented by a “statistical” cor&nation P0)

=

The

hand. a of vectors

F I u&r) > pi < U,(C) I. in which p, is the probability of state 1 z+ >.

WC am interest& in the dynamics of a Primary Subsystem (PS) as it is modulated by the remaining Background

(BG)

always hu written in the fcrrm

Subsystem.

A pure state of the composite system can

170

where

the

In>

(“wavepackets”)

arc

for the BG.

of correlation.

I&)>

=

CC,(r) L

The Liiuville

1CD_>

to he normalie

vqfp(f =

ja

vxxtoxs

but they are not

vary from one 1IL > a, another

simple dinxt

product

is a

are the same, so For maximum

6,.

for the system is W.pl

E

Lp(t)

(2)

I

The difficulty of solving the LE for many degrees of frudom Sinccourfocusison

arc associated

of correlation means that all a_

=

Equation (IE)

G(r) =

are assunxd

The fact that the a_(r)

Absence

I n > *p(r)

cortclation WC have x a_

the F’S and the a-0)

The e_(r)

necessarily mutually orthogonal mea&e

of

vectors

basis

thcdynamicsofvariablcswithin

is generally appnxiati

the PS. we traceovertheBG

t6.81.

subsystem

variables to produce the Reduced Liouvillc Equation (RLE),

where ;ft(o, t) is an effitive

Liouvillian

operator and CFis the reduced density operator fW

thi PS.

Of cow of u(t).

Eq_ (3) is a formal statement and the problem is to determine the smxtum

Note that the RLE is. in gcncral. a m

for the PS. nonlinearity is

intrc~Iuced by the modulational effects of the BG_ Another

approach to the Liouvillt

Equation

is with use of projection

formalism. Let P be a projection supu-opcratoron Liouvik

opcralor

space [12]. It is straightkward

to show that the projected equation of motion has the general fokn , (4)

where Q

=

1 -P.

If P projects diagonal elements, then J5q_(4) is the familiar Zwanzig

equation [ 13). Often. P is defined so that Qp(0) vanish-

and we assume that to be the case

here.

For oti purposes.

WC assurnc that P projects onto diagonal

subsystem and onto some suitable subspacc of the Liouville trace over the BG, we obtain a Projected-Rcduccd

elements for the BG

spaec for the PS.

Liouville equation (PRE)

If we then

of the form

-

171 I

i&t) =

~“(cr,r~t,

-i

$dr*h(cr,t

-I,)

cF(i,,

(5)

0 Sevefalfcanrrwr (iii

ofthePREshouldbanotc&

(i)Thtmoti~eralcas.eisnonliuear~a_

Tha secoud term ou the rhs is a memory term khich arises from the pt-ojedon. (iii>

Physically, the memorytamresultsfxnnthehistoxyof~

cractiouswiththeBGandf?om

the subspace of the PS belonging to Q. Eq(5)isthebasisfora . . _ &uumhm on the other.

fl.uAm&taldichotomy; Diipation

namely,~

outitharidtid

is produced by the memulytcml_inparticula.rinthe

limitthathisHeHnitianandpmportioilalto6(r-t,)

Auim~atcaseresultsinthe

dissipative-linear limit. whereby

i;r = This is the Stochastic Liouvilk

~“-iI-)cT(t)

Equation (SLE)

damping operator- In the SLE$?tends pHxitigaansvGlsc

(

developed by Kubo f14]. iu which r is the

tomaintain coherence; while ITis purely dissipati~

and longitudinal relaxation of noudiagcmal and diagonal elements of

a, respectively_

Ona may fbthcrr;oject the SLE onto thadiagonal &lements of cr and, in the strongly dissipative limit, Eq. (5) leads to a master equation for the probabilities of the PS.

n This is the limit

of ordinary kinetics.

Ou the othn hand, if dissipation is negligible,

the PRE of Eq_ (5) gives rise to the

coherent-nonlinear limit

This is the flu& discussedbelow_

form compatible with the soliton model of Davydov-Car&i-Scott of course_ in

to tm

thalinear limit and for a pure state, Eq. (8) is cquivaltnt to

tha Schrotdinger equation_ We are interest& in systems for which ihc BG is at thamaI equilibrium_

This is the

situation frequently encountered both in biological systems and in molecular spcctroskpy. In particular the initial PS state may be a(O)

=

P=p[-BpHJ

&I

,

172 where

where 8 = (kT)-’ and H is tha system

in which the PS is “coufikd”

equilibrium

accordiug to Eq_ (3. (1)

Hamiltonian_

is a state of consttained thtxmd

o(o)

to a subspace

In general. such a system evolves

Ph enomenologically~ two kinds of situations are widely kuowu:

c

IlIePSisiuitiallyatthermal

equilibrium iu subspace P, but diffiuces over into QI which is also at thermal equilibrium_ Thescantattimetis a(0 where q,#sa(O)

=

P&MO

+

P&t) q

(10)

.

of EQ. (9), CT,is the countcqxut of CT*with P replaad

by Q. and PO(t) and

PI(t) are the time-dependent probabilities, Two examples of equilibrium diflkion

are (a) Kramers theory {l!FJ for a patticle

diffusing out of a potential &ell and (b) Absolute Rate Theoq

of chemkal reactions [lq.

Note. however. that Kramers theory does not require thermaI equiliirium.

merely a gCneral

stoclia&c BG. A simpla physical model of equilibrium flow is the leakage

of a gas through

a pinhole

into a lower pressure chamber at constant T. (2)

WowinPm

Thlsisaagcucralcasaiu

which the rhs of Eq- (5) is complicated by either strong inua-PS coupling (first tetrn) or by strong PS-BG

iutemctious (memoqr tam).

The latter produces nonlinearity if the BG

responds greatly to PS trausitions. so that thermal equilibrium of the BC3is lost and the BG state dtpcnds on the state of the PS. Even in cases where the xzmory iutra’PS.coupliug applicable.

term is stochastic and the BG remairu at equilibrium.

may cause nontquilibrium &low within the PS. Then Eq. (6). the SLE, is

This is a very important situation for many aspects of biomolecular dyuamics

and for molecular spectroscopy. examples. The

COHERENT

Coherent laser-excitation dvlzpmics off&~

many good

formerwilt he dis4zussediu the next section_ MACROMOLECULAR

Tha field of tkchckistry biochemical reactions

DYNAMICS

has traditionally been concerned with the energetics of

Standard thermodynamic quantities, such as AC’. AHo. and AS’ arc

dctcrminadandcatalogcdfortach~epofcvtry~tabolicpathway. how much useful energy is available at a given temperature, T_

Thtpurposcistokuow

173. Along The

with

kinadcs

encrgcdcs,

staudard approach

tif tnzymk catalyzed reactiouk is also ti central f-

is Absolute

FZate Thbory.

whereby

the smudard

activation

thermodyuamicquantitiesalzmtao~ The important point here is that our concepts of both biacrkr@cs based on a picture in which the system is at thermal equilibrium accordkg

to rho equilibrium

diflisifm

and kinetics arc

and processes occur

limit discussed in this pmccdiug section-

In this

section, we shall develop an aIteruativc pictures One of the ky mechanisms whereby cucrgy is trausfa-rcd in bioIogical systems is by coupled reactions. For example. the syuthcsis of acctyl coenzymc A from acetate and CoA rcquirts energy which is providedby

the hydrolysis

of ATP.

Since the mechanism iuvolves

the intermediate fonnati on of acctyI adcnylatc. the reactions arc _ sense that ATP is not hydrolyzed in the abkncc as

--:LI~

in the

of acetate

On the other hand. there is another class of coupled reactions which we T&& to here “” e These involve the intermediate role of a macromolecule (usually

a protein) in the transfei of energy. The idea is that two subsystems (reactions) arc coupled via a macromolccula in.a manner that the energetically “downhill? k&on in the absence of the other reaction membranes which rcque

could proceed

Good examples cau be found in ion uansport ackss

the hydrolysis of ATP_

Two important features of noncsscutial couplihg arc: (I) one subsystem relaxes toward equilibrium,

while the other is driven away from equilibrium;

mediated by a kcromokcule,

which operates in cycles. This is analogous to photochemical

processes in whkh one molecule absorbs a then undergoes reaction

and (ii) the coupling is

and transfers the euu-gy to another, which

(The first molecule is the intermediary_)

Let us focus upon the macromolecule, denoted by X During the course of the mtion X is activated by the driving subsystem and is deactivated by the driven subsystem:

x0x-

,

where x’ represents a set of one or more act&cd Macromolcculc

Activation (MAC)

Cycle.

Some relevant questions arc as follows: cycles most efficiently?

(excited) states. We refer to this as a

(1)

How cau biological systems use MAC

(2) What is the thermodynamic nature of X?; Le.. to what extent

is x’ a quantum mechanical stationary, thermally equilibriated state? (3) ckctronic-conformntIonal

structure of r;

We assume at the outset that MAC

What is the

how many states (species) are involved? cycles operate as p

*

Cocsider the energy-entropy plane. in which energy. E. is thu ordinate and entropy, S. is the abscksa

A hypothetical MAC

cycle might appear ofi the ES

plant as follows:

174

Xl0 and &- dcnotc the ground (unactivated) and two activated thumodynamic

Here &

states. The slope of the dashed lIns is the temperature, T. To understand the thermodynamics of tha MAC

cycle. we begin with the fuudamcntal

relntion,

dE

=

Tds

+

FF,d&

C11)

,

wh&e the last term represents the contribution of all mechanical and “compositional” (e.g.. chemical potential) forces, F* acting on the subsystem’s extensive variables- For the special case of c~n~fanf T. WC have

for the Hchnholtz potential (fke

cuergy),

k

This leads lo the well-known

constant T

relation A& for each step of the MAC

TAS

=

-I-

AA

(13)

,

cycle-

Note that the dashed lint in the Es

plant is also a line of constant k

the left of the dashed line have AA > 0 relative to state X,

The value of AA for

All points to

the transition

from X,, to a point representing X; is given by the vertical distance from that point k the constant A (dashed) line for X, According to the first law of th&modynamics

for a closed subsystem, we have

(14)

AE=q+w, and for the second law, at constant T,

qSTA.5

a2M

,

(15)

f&r th& heat absorbed by. and the work done on. the subsystem Now consider the MACcyclc

depictedabove.

The activation stepX, +Xi

has AA > 0.

‘.

176

s.oworkm~~bedoneonJC.

Howcvcr.both~~~X;andX;~X,havchA<.O,socach_

may occur spontaneously with a possibility of work lxing done by X on a.nothCrsubsystemForeachstep.AEandASar&mzero_ At first sight. this appears m be an ideal system for mrnsfarring w&k potential from one subsystem m another. But. there is the foIlowing which~cOrcquircd(2dlaw)lhatq
cave&z Each of the (two) steps fm one might

Thcserepresentheatlosses.

argucthatthc-hcatl~insocpo~X,~X~.~comptnsatedinmtptwo.X~~X~.f~which AS > 0, However* it ti

not follow namssarilythatq~Oforsacptwo,infac~qhamtha of maximum work efficiency_ In reality. it may have

maximum value TAS only for t&case any value less than that.

BothstopZandstcp3artspontancous;ic,McO,sothey~norn~ydrivul The question is not whether they occur spontaneously, but whether they tend m occur with maximum WbXc

transfer of work potential m another subsystem.

<0,soncccssarilyq<0,whichmcans Thug

thatpartofAEislostashea.ttransfizrmtheB~-

even for maximum efficiency,

limited_

Consequently.

Cixxidcx step 3: AS

work potential tmnsfer (m another subsystem)

tha ovcmll efficiencyof the MAC

is

cycle hinges gristly on the

efficiency ofstep2,

We present the argument that. in (closed) nxural sy4tma. wbichAS~Otendmoccurwith minimumworkefficiency!

isothermal proaxes

maximum spontaneity (irreversibility) ar& consequently. Machanismsof(~hanical)couplingwhichpmvi&meximum

work potential transfer from X should involve AE < 0. rather than AS > a m our &uppusiti&. it would appear that MAC be relatively inefficient

for

This brings us m the following

e #1:

Thus. accord@

cycles having large AS values would tend m

For proaxes

importan t point of summary.

in which tran&r

&f work potential f&n

one subsystem to another is an essential factor. Nature tends to use MAC

cycles for which

the individual steps have minimum [ AS 1 values. What about the s&xure

and dynamics of X? It is frequently a protein molecule or

an organized aggregate of such molecul&_ low-energy

Usually. X,, is the ground clecuom -c state fora

conformation; and the Xrg am confonnational

configuration space Because

of the

exci&ons

m nearby r&ions

presence of many local minima [ 171. it is

many Of the state8 of X,O are nearly degenerate.

of

expected that

In the them%cdynamiC limit, each Xr* is a

thermodynamic component whose properties are goverued by its partition fbnction. quantum states of each are statistically independent.

The

Thus, if X,* has N,quasi-degenerate

states. there is a contribution m thu entropy given by S,* = k In N1. The associated loss in quantum mechanical phase can greatly affect the dynamics (kinetics) of the system. as was discuss4

in the previous section (fa

example

by dilution cfftcts)_

176

On the other hand if the transitionsof X arc quantum-mechanicallycoherent, there is a one-to-one correspondencebetween the states of X, and the preparedstates of X,,; the lattcra+ikc doorwaystatcsofspcctroscopy W]_

-rhccorrespon43ingcntropyofactivation

iszero- Thisbringsustoaftmhcrpointofsummary. MAC

The entropy incmmcnts of MAC cycles arc minimized

if the preparedstate of the X,* arc coherent (nonstationary)quantum staocs. _.

Accordingly. thc~ -of

I

.

of quantummechanicsreplacesthe-

statisticalmechanics. Possible advan~s

to the biological

system arc (a) high cfftciency of transferof work potenti& (b) enhancedkinetics (absence of dilution cff&cts, and (c) temporalco&-di~ticr_ sf p_roccsses.

SQLlT0NMEcEMNIsn-i An example of coherent macromolecular dynamics which fits the criteriaof a MAC cycle is the a-helix soliton model introducedby Davydov [9] and modified by Carcri [IO]. et al.. and Scott [ 111, The idea is thatchemical energy (e.g., energy of hydrolysis of ATE”) can be propagatedalong the *helix from one site to anotherby means of a coherent soliton wavepacket. Thi soliton consists of an amide vibration(mainly carbonyl) which propagates as a coherent txciton whose dynamics is modulated by a certain kind of background BG) mcxkzbelonging to the a-helin

For the Davydov mode] the BG mode is a l&ration of the

amide group itself; whereas,for the Car&/Scott model it is low-fi-equcncyvibrationof the H-bond pmronof thea-helix. In tither case, accordingto the solitonmodel, theBG coupling introducesa no&car

term into the Schrocdingcr equationfor the cxciton_

The basis for the model is that the BG mode moves in a Potentialwhich depends on the stateof the carbonyl vibration, Thus. excitation of a given cat-bony1changes the force on the BG mode which tends to displace its coordinate. Thii is an example of adiabatic modulation which we have used extensively in other aspects of subsystem modulation dynamics (5-7 ]_ The Hamiltonian for the system has the simple form H

=

X I

[H,oqymb;b,,+V

(b,‘,,b,+-b,‘_,bJl

s

(16)

Here H.’ is the Hamiltonian for the BG (proton) mode of amide group n, y. is the proton coordinate,b,’ and 4, arc creationand annihiIationoperatorsfor the amide v%ration x is a coupling parameter,and V is the excitation transfercoupling for neighboring amides. WC

Thccliticalr4ssumptionofthcthowyistbcAnsatt:)u(t)> that the systan

=

I ~A0

) I *,@I

r, n&y

sta@ is quantum mechanically ~~~asaproductofanexkz+rt

audaphonon(protonmodc)part

Bytxpandingthecxcitonpautiuthcamideproductbasii

wehave

Iu(r)>

X I n > C,W I *,@I >

=

m

I

KY

next step is fo f
for the cxciton amplitudes as

follows: (19)

ASCC&importan

t assumptiou is that the time-dependent effkctivc Hamiltiad

for the

cxciton. resulting from FQ_ (19). can be averaged over the period cf the proton vibration_

By doing this and using m_ (18) ~JOget w

iC,

=

which is the nonlink isthtfonx

pc, +

1 cm

-4-y

V(C.+,

f

G-1)

m

Schrmdingcr equation for the cxciton (soliton) amplituk

HUG

k

cDnslantfor~pmtoumode,

Eq_ (20) has been formulated and calculated extunsively by Guai. [ 113,anclLomdahl[19].

Computer

simulations

produccsoliton-like wave

We have recently carried our a rigorous fmulation of motion. JZq_ (l?).

-&ich

Scott

packet dynamics.

and calculation of the equation

avoids the soliton Ausatz. FJ.q.(IS),

procedure needs3 u) produce Eq_ (20).

et al. [lo].

and tit time-averaging

Our purpose is to evaluate the integrated form of

Es- (17).

lW:r)> for various initial state I &B somewhere iu the polymer.

=

exp(+w

I yo ’

in which a single ami&

*

WI

vibrational excitation is localized

Tha Hamiltonian is that given by J2q.(16).

1.78 A tractable fcam of the timc-cvoXution opaator

coordinatadi~l~

In Eq_ (213 can bt

topera- fortheproton mofdhatts,

fcmmd by using

y.: namely. DZypm

= 3%+ d,,

whau =

D. HCXCp,iStht

=P

(-h#,)

[RI= l]

momentum operator conjugate

to Yp’

By

(22)

I

using a

harmonic m-1

for the

proton mcxlcs.

H,” = it follows, with d, = k”fizb_.

(2ml-W +

that D+(H” -C+D~Dl]D

=

H

D

=

HO 6

c-1

.

lmY,l

=

5

=

I-I

D,

,

k I

H,”

.

V@:+,b,

=

c

,

+

b:-,W

.

1/2k-y

The linear potential term iu y, has been shiitcd to the cxciton term_ The equation of motion can thus be written

It&)> wi*h H

=

p

-C

wh&c s* Ad &

1u&B

=

=P

I%& (a,*--a.3

,

for y modes of mass m and frequency (10.

Eq. (25) provides the folbwing

physicti picture: At t c 0, the p&ytxr

state (no ami& excitations and no y-mode photins). propagates at amide n until the excitation is uansfm

is in its ground

Amide n is excited at t = 0. at which

point D disppIaccs the ground state of phonon made y,. D+ shii

(23

,

3p src the creation and annihilation operators for the ym- mode quanta and

(Zmti?)-‘3_

=

up(-izrp

It is helpful to express D, in the form

c D CD+.

0,

D’

=

This displaced (coherent) statu

to, say, amide (n + 1). at which point

the y. state “backward” and D shifts the ya+, ground state “forward”_ This shifting

b&k&dforth

occursbecauscwrcimtion~nsfcrproduccsachan~inforocontheas~~

y-modes. themby creating phonon excitations (via the opxators

a, and &’ in Eq. (26).

179 We have devclopcd

ktnalgorithm

for solving Eq. (25) by a noupertmb~dou method

[18]; It COII&~S of first transforming to the interaction picture in H”, which eliminates the @ in Eq. (25)_and installs a, exp (-ia

and h* exp tjcut) in place of a, and q,+ iu Eq_ (26)_

The resulting V(r) is expanded in a series

WC have shown that only the k = 0 term makes a significant contribution. leading to I do>=

wJwv)I

%>

cm

-

Computer codes were developed for the Cyber 205 Supercomputer and extensive calculations of E.q_ (28) were made for various properties associated with exciton motion in the a-helix [IS]_ Thedetails of the method and theresults show unequivocally that soliton motion does not hold for the a-helix

Dispt&m

interactions which cause the &x&on to

spread greatly as it migrates dominate this motion. This is a-ue for a broad range of values for all parameters of the model: x. CO,IL and I (expressed in units relative to V)_ The reason for the failure of the soliton nxxlcl appears to lie in the Ansatz. Eq. (18). that the siats evolves & a simpIe product. which quantum mechanicalIy is un&rrelated. Subsystem intemca*ens naturally

produce

and our results show that they are not

correlations

negligible- In fact, in arelated shady Brown, et al. [20] have shown that a product ofcohc xnt states ests variables.

in time only if H is a homogetius how,

results of &own,

function of tot& degree one in the action

the two components of Fq_ (18) are not ~uir&to tt al., do show

the severe

consmints

he coherent, but the

placed on H when conditions

are

placed on the m-ucture of the state vector_ CONCLUDING

REMARKS

In this paper. we have discus&d a number of similarities cnzountered in the study OF molccutar spectro&opic processes and of biomolecular dynamics_ Both of these areas have been instrumental in shaping our own approach to subsystem rn&lulatio~ dynamics and the as&at&

subsystem equations of motion. OF particular interest is the distinction between

equilibrium diffusion and nonequilibrium flow in phase (Hilbert) space_ The most important feature of this paper is the discussion of the fundamcnml

role of

cyclic l&ocessts involving biological n&ct-oruolccul~_ We refer to the& as macrom 0tecuIe activation invoIvcd.

(MAC)

cycles and present conjectures

on the nature

of the “excited”

states

The solit& model of Davydov. and modified by Carti and Scott, came to our attention some time ago as a possible mechanism ofcoherent local&d an example

of nonequilibrium

Schroedinger &ation,

flow.

energy transfer. representing

A careful formulation

without the soliton Artsa-

and calculation

of tbt

shows that the soliton mechanism is not

opcrativc rather. the excitation migrates as disptrsing cxciton waves. This is adisappointing result, since the soliton mechanism

is such an attractivu concept.

Ncvu-tbeless.

originators of the soliton model have made a contribution ofheuristic value of c&hcrent and conccrtcd motion in biomohxular

the

forfuturestudies

systems.

ACKNOWLEDGEMENT Thii work was supported in part by Contract No. DE-FGO586ER60473 Division of Biomedical and Environm ental Research of the Dquutment

between the of Energy and

Florida State University. REFERENCES 1. 2. 3. 4. 6 7. 8.

9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.

G. Car&, A. Giansanti, and Rupley, Phys. Rev. A, 37 (1938) 2703; Proc. NatJ. Acad. Sci, U.S., 83 (1986) 6810. ~$FFodcs. in Radiationless Transitions, cd. by S-Et Lin. Academic Press. New York, W. Rhodes. J. Phys. Chcm, 87 (1983) 30. A.E. Douglas. J. Chcm Phys. 45 (1966) 1007. W. Rhodes, J. Phys. Chcm, 86 (1982) 2657. W. Rhodes and A. Velenik. in Mathematics and Corn utational Concepts in Cbcmistry. cd by N. Trinajstic. Ellis Horwood, Cbichcstcr~ 19 E6. W_ Rhodes, in mth-Chcm-Camp, cd. by RC Lather. Elscvicr. Am 1988. P. Grigolini, in Memory Function Approaches to Stochastic Problems in Condcnscd Matter. Adv. in Chcm Phys. Vol. 62.. cd. by M.W. Evans, P. Grigolini. and G_ Pastori Parravicini, Wiley and Sons, New York, 1985. AS. Davydov. Solitons in Molecular Systems. Reidel, Dordrccht, 1985. G. Car&; U. Buontcm , F. Gallu4 A-C Scott, E. Gratton. and E. Shyamsundcr, Phys. Rev. B, 30 (1984 p”4689. AC. Scott, in Nonlinear Electrodynamics in Biological Systems. cd. by W.R Adty and A.F. Lawrence, Plenum, New York, 1984. RR Ernst, G. Bodtnhauscn. and A. Wokaun, Principles of Nuclear Magnetic Rctinancc in One and Two Dimensions, Clru-cndon. Oxford, 1987, R Zwanzig. in Lectures in Theoretical Physics. Vol. 3, cd. by W. B&tin, B. Downs. and J. Downs, Interscience. New York, (1961). R Kubo, Adv. in Chcm Phys.. 16 (1969) 101. HA Kramer-s. Physica. 7 (1940) 284_ B-H Mahan. J. Chcm Ed, 51(1974) 709. D. Poland and HA. Schcraga, Theory of Helix-Coil Transitions in Biopolymers. Academic, New York, 1970. A. Nicholls arxl W_ Rhodes, to be published_ PAS_Lomhdahl, in Nonlinear Ekcuodynamics in Biological Systems. cd. by W.R and A.F. Lawrence. Plenum. New York, 1984. D_% Brown. K. Lindenberg. and B;J_ West, in Nonlinearity in Coadcnscd Matter Physics, cd. by A.R Bishop. et al.. Springer-Vcrlag. New York. 1987.