Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://library.biophys.msu.ru/PDF/3331.pdf Äàòà èçìåíåíèÿ: Fri Mar 14 17:51:52 2008 Äàòà èíäåêñèðîâàíèÿ: Mon Oct 1 19:32:06 2012 Êîäèðîâêà:

(

Vw ,. R"bilL , .. C/lt li.,b 1",,..,1,,1 ,,, C' IpJ'n'II},' " 1001 b.rMA I I< · ,.,. ..ts,II"'Ofl't', ,, ,J" ',, " / R... u J, ;, " w B·

.,.,.vsIa.

O ,;¥u.s.1 R...,,,,, ,, Tao'

CE L L IlIOI' IIYS I CS

= === = = === = = ==

New Sp a tiot em p oral M odes in the R ca cti on- El ectrodiffu sion System
T . Yu. P tyu snt na -, A . I. Lob anov - , A. I. Lavroval, T . K. Starczhi lc va- , G . Yu. Rlznic he nko -, an d A. B. R ub in!
IBio lo g i ca l Facu lty, Moscow S ta t e Universit y , M osco w , 119899 R uss ia lMos co w Institu t e of Ph ys ic s and Tec hno lo gy , D ol goprudn y i. Mos cow Re gi on. /4/700 Russia

Received November 26, 200 1

Ab st rac t-s-A st ud y w as mad e of the prope rties of a rc ac uo n-elcctro diff usio n system . A two -co m pone n t mode l was dev e loped to d escr ibe how interac ti ng c ha rg ed pa ni cles d iffuse near the membrane in

low -ionic-strength media for which the common assumption o f electroneutrality is invalid. Anal ysis of this rnodel-c-ccnsuucted to take into account the presence a self-consistent field-show s that the latter contributes to the emergence of bistability, localized structures with highly heterogeneous spatial d istributions of charges, and spatially and tempor ally aperiodic modes.
K ey words: elecuodiffusion, self-consistent field. nonlinear dynamic modes

INTRODUCfIO N
In vol ve ment of charged molecule s in many c he mical a nd bio l og ic al proces ses leads to the need to examine the role o f the so-called self-consiste nt field, or e lectric field arising fro m interact ion of mo ving charged particle s, in p rod uci ng various d y na mic mode s. Io ns are essential for live o rganisms. Among ionic processe s i n the ce ll. most importa nt are t he ge nera tion o f potenti al grad ie nts across the memb rane a nd the move me nt o f e lectric p ulse s. Th ere are plant specie s that fonn a lternat ing zone s of high and l ow co nce ntratio ns of so me ion (i n te rms of nonli nea r d yna mics. such zo nes are k no w n as di ssipati ve str ucture s). Fo r exa mple, ac id ic and al kal ine reg io ns a lternate o ver the surface of filamentou s algae Ni te lla and Chara [ 1-3]. Th ey d iffer in the membra ne pot ential [4J a nd the po ten tial of t he adj ac e nt laye r (5 ]. Thes e pote ntia l vari ations are a source of add itional e lect ric grad ie nts along the me mb rane . To st ud y the mechanisms o f such phe no men a . it may be esse n tia l not o nly to kno w ho w parti cular ion c hannels and memb ra ne c arri e rs ope ra te, b ut a lso to u nderstand ho w the d y nam ica l sys te m beh ave s a s a w hole o r, in othe r words. ho w the spatiot e mpora l organiza tion o f e vents is affec ted by the pre sence o f a self-c onsistent field .

T he re are t wo approac hes to d e sc ribing elec t ric p he nome na at an d near the membra ne (6 ]. One ap proa ch called the equ ivalent electric circuit technique was e mployed in st udies o f nerve imp ulse condu ct ion [7] and of potentia l profi le s in Chura ce lls [5, 8J. The seco nd a pp roach is to use the elec t rodiffusion eq uations, as in st ud ie s o f the e ffec t o f e xte rna l el ectric fields o n chemical reaction s in hlgh-i onic -strcn grh solution s [ 9- 13] , fo r which t he el ec u o ne utrali ty co nd ition ho lds. Thi s co nd itio n, if fulfilled . mar kedly simplifie s t he a nalysis of proble ms. Ho we ver, in bio logical media , unlike ch emical media , t his is o fte n not the c ase. For exam ple, de part ure fro m el ectr o neu tra lity is possi ble near ce ll membra nes as a result of active ion transpo rt . t he pre sence of t he e lectri c d ou ble layer , and the prese nce o f fixed charge s o n integral memb ra ne protein s. T he goal o f t his stud y was to de monstr ate that sel f-con siste nt fie ld -rel a ted red istributi o n o f memb rane charge s and potentia l gradie nts gives rise to a n umbe r o f importa nt biolog ical effect s in sys tems with nonl inea r c he mical kine tics. We attai ned this goal by so lvi ng react ion-electrodiff usion eq uatio ns for lo w-ionic-streng th media whe re the c o mmon assu mptio n of electrcneutrality is invalid .

266


REACfIO N-ELECfRODlFFU S ION S YSTEM

RE ACT IO N-Ef,.E CT RO Dl FFUS ION MO DE L

Fo r m u la ti ng t he Problem Le t us co ns id er a si ng le ce ll p laced in a we a k e lec troly te so lu t io n. Th e ce lJ me mbran e i s kn o wn to carry an e lec tri c c h arg e . A ss ume tha t its d is tri b ution ove r the c e ll s urface is un if orm . T o sc ree n t he ce ll s urfac e ch ar ge , elec tro l yte io ns of o pposi te c har g e a cc u m ulate nea r t he ce ll m e mbr a ne [1 4 ]. T he sp an o f t h is oppos ite ly c harg ed laye r c an be es ti ma ted : it i s the Debye radi u s (le ngt h ). Th e Debye ra d iu s i s us u a lly m uc h s ma ll er th an t he ce ll di ame ter. The re fo re , t o mod e l th e e ffec ts p roduced b y io n mo vement a lon g t he ce ll mem bra ne, we c a n c on s ide r th e l atte r as a n i nfi nite bo und a ry c arryin g an e lectri c cha rg e . Thi s ass u mp tio n all ow s u s to g o o ve r t o a o ne -d ime nsio na l prob lem .
I n the sys te m o f in tere st , all rea c tio n s a lte ring t he con c en trati on s o f c h arg ed parti cl es pr oc eed on th e me mb ra ne s u rfac e . Th e e lec tric d oubl e la yer is su ppo sed to be in eq uili b ri u m w ith th e rest o f the e lec tro ly te : th e flux o f ion s lea vin g th e lay er by virt ue o f their th erm a l m oti on eq uals the flux of io ns arrivi ng from the bul k s o l ut ion. M alh em a ti cal M od el Le t the sys te m und e r s t udy c o ntai n neu tra l p arti cle s a lo ng w ith c h arged o ne s . C h ar ged particl e s m ay be io n s d iff u s i ng a lon g th e ce ll m embrane a nd tak in g part in var io us c he m ica l re act ion s . Let th e co nce ntra t io n of ion s be m uch lo we r than that o f ne utr al mo lec ules. I n ot he r wo rd s, we c on s ider a lo w- io n ics t reng th so lu t io n . Var iati on s in ion c o nce ntra tions are de scr ibed by th e rea c t io n-elec t rodiffusio n equation s [1 3]. With t he c o nce nt rat ions o f p o s iti ve and negat ive ions des ig na ted ' . a nd ' 2. re sp ecti vel y , th e set of dim ensio n le ss eq uatio ns with o ne s p at ia l var i ab le (r) read s

where 't is ti me: ftcJ ' c2) and g (c], C2) a re n on linear fun ct ion s t hat d es cri be t he c hang e s in io n co ncent ratio ns c ause d by c he mic al react io n s at th e me mbra ne : D . and D 2 a re the d if f us io n coeffic ie nts fo r pos it ive a nd ne gativ e io ns, re spectively ; B. and B 2 ar e th eir m o bi li tie s in an e lec t ric fi e ld ; y is the ra ti o of the char acte ris tic ion co nce ntra ti o ns ; and z i s th e va lence ratio of ion s . Th e equ at ions de scribin g c han g es in io n co nce nt ra t ion s we re so l ved fo r n o- flu x b ound ary co ndi tions :
J , ( O 1) =J , ( l, 1) ,
1,(0,1)

=0
~

=J

, (I , 1 )

( 2 . 1)

O.

As th e ini tia l co nd itio n, we chose th e steady sta te un ifo rm di s tr ibuti o n of io n conc en tratio ns, tha t is, t he so l utio n to th e alge brai c s et j{c], C2) = 0 and g{e " e,) = O . Th e bo undary co ndi tio ns for t he P oi ss on equat ion were ta ken in t he fo rm
IjI {O, 1) = 1jI{I, 1) = 0, ( 2. 2 )

impl yi ng th e a bse nc e o f elec tric fiel d s p ar a llel to [he mem b rane s urfa ce.
A N AL YS IS FO R SE LF - S IM I L A R IT Y

As s ho wn b y O n cle ve an d Sc h mid t [ 15} . th e low er t he io nic stre ng th , the grea te r the e f fect of a self-co nsis t en t fi eld . T o bett er und er st a nd t he pat tern in g in lo w- ioni c- stre ngth med ia , le t u s co ns id er se l fs i m ila r so l uti o ns to the s et o f equa ti on s ( 1.1 )- ( 1.3 ). Th i s appr oa c h allow s us to take ac count of t he qu adra tic te r ms d e sc rib ing a s e lf- co ns is te n t fie ld with out t he need t o as s u me th at t he mediu m is el ec tro neutr al. It is kn o wn th at , if a s et of p art ial di ffere ntia l e q uat io ns ad m its a sel f - simila r so lut ion , the deg ree o f t ha t se t can be redu ced . F o r exa mp le , it is p o ss ible to go ove r t o a s et o f o rd in ary differential eq uat io ns [ 16]. As s uc h sys t em s are chara c te rized by self- s im ilari ty [1 61, t he s o lut io ns to the s et o f ord inary di f fe renti a l equ atio ns w ill re flect the so l ut io ns to t h e o rig inal se t, p ro vided th ai large t ime inte rv al s ar e co ns ide red . By s el f -si mi la r so luti o n s we m ea n the so lut io ns ob ta i ned b y a ge nera li za t io n o f th e v ariable sepa ra t ion tech niq ue [17 , 18 ), acco rdi ng t o w h ich the so lu tion of t he orig ina l eq uat io n is w ritt en as F t r . "t) = Cj) ("t ) h (~ ) , w here the seco nd m ultipl ier depe nd s only o n som e com bi n atio n ; (r. t ) o f in de pend en t d ime nsionless variab les .

dq = D ( . + H dq 12,

Crt

J

ar

2

J

ar ar

a't' _
( 1.1 )

- ~ X q ( CI - tYc 2 ) + !(q ,C2),

0' 2 =D
Ot
- B2

a2C2 _ 2

ar 2

ckr2 a;a", a

( 1. 2)

+B2 XC2 ( C.

- ZYC2 ) + g ( CJ ' C2 )'
-zye, ~
2002

a'ljI =- x(e, ar'
BIOPHYSICS Vol. 47

(1.3)

No.2


;

PLYUSNlNA e t aL

Y,

Let u( r , 't ) and Y(r , 't ) be smal l d e viatio ns from the val ues of Cl and c2 co rrespondin g to the unif orm steady state (c? and c ~ . respectively ). S elf-similar variab les exi st i n the syste m (1. I H1. 3 ) if its "c he mica l" part (equations describing the ch e mic a l reactio ns) contains o n ly quadra tic term s ( t ha t is. if l inear te rms and te rm s of the third an d higher o rde rs in t hese eq uatio ns are neglig ibly small ).
Expa ndi ng t he rig h t-side f uncti ons in power s o f t he new variables u and \I. a nd re tai ni n g o nly q uadr atic terms . we ob tai n

Fig . 1. Modulated structures , as ca lc ulated in t he se l f-siml lar syStem (S.l H 5.3) fo r D IJD, E 8 ,1 2 = 3. 8 (Ill " a,P1 2· 14, P 13 · I, P2 "' 0.5,P22 :: - 24 . (ln = 2.85, 1

:-=I, y .. I , I OOX - O.99.

a u
at

= DI

a2 u ar2

+B1 a r a;+PIIU + Pl2u v + PI) \! ·
( 3 . 1)

a aw u

2

2

Y,

( 3.2)

,(
·
FIr.. 2. Lccaliaed struCtures with high s pa tia l va ria t io n in

-=-x (u -ryv ).

a' ljI

ar'

( 3 .3)

charg e d ens i ty. as calcu lated in lhe sc: lr-similat syste m (S.l H S.3) for Di ll)". B I/S, "" 2.6, Pll = O P l2 " - 25. . P ll "" 1.P21 . Oji . Pn - - 24. Pn· 2 . 85.1: = 1. 1 = I . a nd X .. 1.035 .

w here c onsta nts Pi; are c ombi natio ns o f the second -order rate co nsta nts o f the "c hem ical" part o f the syste m a nd of the parameters of its "el ectri c " part. The Pu c an take on positive an d negativ e va lues . The self-s imilar so l ution will be sought in the fo rm
u( r . t ) = -Y, ( ~ ).
'r

,

I

, ( r . t ) = - y , (~ ).

I

t

(4 )

Y2. and Y) are the so lu ti o n to the F unctions fo llowin g set of ord inary different ial equ ation s

y,.

DIY''~ YI'BIYI" +'2 + Y3
' ~' D2Y2 +2' Y2 - B

+
( 5 . 1)

+ Yl + Pl IYl2 + Pl 2Y1Y 2 + Pn Y ~ = O.

" 2ZY2Y ) + Y 2 +

o

g u

+ PIIYl2 + P I2Y1Y2 + PIl Y? =

o .

( 5. 2 )
( 5 .3 )

y;

+X (Y, - tlY , ) = O .
._ = 0 . y , I ~ ~ . _ =0.

F Ir.. J. Ph ase po nra.iu of the m ode l descr ib ing the loc a l c he mical dy namics (I) without and ( 2, J ) with n: gOlr d for

with the bounda ry cond itions

the e ffec t o f. KI(-eons islcnl el ectri c fiel d : (I ) st ab le fo -

yd...~

( 6)

cus, (2) saddle. and (1) stable node. Parameters used in c a lculat io ns ; a _S. b.2S, 8 1 - 0.0 1, 8 2 .. 0.01. z .. 1.
., - I, a nd

x. - 1.

Note that primes indicate dif(er ent iation with reo spee r to the new i ndepende nt variab le ~ .
BIOPHYSICS Vol. 41 No.2 2002


REACD ON-ELECTR ODIFFU S ION SYSTEM

C onditions (6) i mp ly that we lo ok for locali zed s truc t ur es . Fa r o f f th e c oo rd inate o rigi n . 't he s ys te m re ma i ns i n a s pa tia lly h o mo g e n eou s steady s tate . The so lutio n can be con tin ued in to the ra nge o f negative ~ val ues as an even fun ct ion b y s e t t ing sy mmetry conditio n s a t ~ = 0 :

5.2"
5.0 4.8 15 13 , II 0.0
0. 2 0.4

O;I oo(l Y ~~

=0

·

~ ~f, -o

=0

,

0 .6

·

;=1 . 2.

Alth ough we c onfi ned o urs elv es t o s ee ki ng o nly s e l f-si mi l ar solutions t o the full model, th is app roach allo ws u s to understan d s ome gen eral ru le s of patt e rn

Fig . 4. St ructures ..... ith modu lated am p lit ude . 3 1 calcu lat ed in the rull IYl lem o r pan ial d iff ere nt ial eq uaucns I I.l H I.3) Icr GI = 5, b :: 25.0 ,:: 10--', O2 = 1.5 ·10-". B , - 0. 1. 8 2", 0 . 15. : - 1.1" I , and X = 0.99.

formation under the action of a self-consistent field in, rea cu on-electrodi ff usi on s ys te ms . Of co ur se, in real
sys t e ms the roles of linear term s a nd te rm s o f the th ird and h ighe r o rde rs cannot be ne glected . Ex pectedl y , t he ir co ntri b u t io n wo uld lea d eith e r to pa uem s tab ilizat ion. or t o s ignifican t d e viati o n fr om se l f-s imi lar soluti on s. Still we s up po se that r ea l sys te ms ma y ha ve initial c ond iti o ns requi red for brin g in g into pla y scenar io s cl ose to se lf- s imilar on es . N ume rica l ana ly s is of lhe se t o f s el f- s i m ila r eq ua tio ns (5 .JH S.3 ) r e vea l ed a nu mber o f mod es . th e se t hat arc a na log o us to t he De scribed be low mod e s fo und i n the fu ll mod e l ( 1.1)-( 1.3). No nmon oton ic s tru c tu res deve lo p in mod e l (5 . 1) -( 5. 3) fro m spec ia l ini ti al co nd iti on s c alled th e C auchy da ta . Th e am pli t udes o f the s truct u ral e le me nts depend on th e s el f-s im il ar var iable ~. With incr ea s in g ~. the a mp litu de fir st p as se s thr ou gh a m a ximum a nd the n le ve ls o ff ( Fig . I ) .

6 .0

1.0

30 2 0 .0

r

are

Fig . S. Localized stru c tu res wi th h ig h spa tial var iat ion in c harge de nsi ty, as ca lcu laled in th e futl sy srem or partla! d i fferenti al equations ( 1. 1H I J ) Icr GI = 5 .21, b :: 2 7. 144, 0, _ 1Q"'4 . 0, _ 1.5· 1Q"'4, B ,:Ii: 0 .03 . 8 2 ", 0.045. Z = I .

'Y'" l. andX - 1.

4.0
2.0 50

Fi gur e 2 s ho w s m or e i nte re s t ing so lut io ns to model (5 . 1)-( 5 .3) : a loca l ized s tr uc tu re w ith hig h sp atia l vari a t ion in th e c harge dis trib utio n . Its s ize is s mall com p a red wilh the s p a ti al sca le o f the pr ob lem . Be yon d the r e gi on occ u p ied by t hi s s t ruc tu re , ion s remain u nif o r m ly d istr ib uted .

49 ,

48

41

46

0.2

r

RESULTS O F NUMER ICAL EXPERIMENTS
Analy sis of t he fu ll sys te m i n parti al d e r ivati ves ( I.IH I. 3 ) shows th a t the interpla y between pa rt ic le tran sfer in a n elec tri c fie ld and d iffusi on gives r is e t o a va ri e ty o f mod es. To h a ve no nlinear te rm s in d es cribin g c he mica l r ea ct io ns. we chose equati o ns o f t he B ru s sel aror m odel :
! (u.v ) =a - ( b + l )u +u~ v ,
BIOPHYSICS

Fig . 6. S tep -hke di l tribul io n o r io n ecn cemrario ns , ali c a lc ul ated in l he rulll )'s tem o r pan ia l d iffe rent ia l equ alio n s ( 1.1)-( 1.3 ) rc r GI '" 5. b :Ii: 25 , 0 1 '" O 2 '" 10--'. 8 , =

_ 8 , _ 0 . I . z" 1. 1_ 1. ;lnd X . I.

g(u, v ) =bu _ u 2 v.

Note th a t the Bru s sela to r is expanded b y incorporating the quadratic ter ms whe reby th e pre s e nce o f a se l f-c o ns i st e nt field is t ak en in to c on sid eration . In the ex panded model . more stead y s ta te s are poss ib le than i n the ori gin a l o ne (Fi g . 3 )_ Speci fically . i n

ver ,

47

No. 2

2002


270

PLY USNlNA et a l.

add ition to th e s t ab le foc us (poi nt J in Fig . 3) co rre spond ing to the initia l stead y sta te of the Brusscl utor. t wo mo re eq uilibrium po i nt s e me rge , that is . a sa d d le and a stable node (poi nts 2 and 3 i n Fig . 3). The el ectron eutrality c ond iti on is satisfied o n ly in s tate I . Thus . taking in to acco un t the presence of a se l fco nsisten t field, we find that the stable state in wh ich th e c o nd it io n of elect rone utrallry hold s c oe x is ts w ith the states in w h ich the charg e ( nega tive in t his mode l ) is not co mpe nsated fo r.

Let us exami ne th e effect of a se l f-co nsis te nt fie ld o n the mod el beh av ior for the pa rameter va lues a t wh ich the sys te m is Turi ng-stab le. Th e ini t ial co nditi o ns w i ll be sma ll devi at i on s from the e lec t ro-

Th e eme rge nce of addi tio na l ste ad y states in whic h the elec t rone utrali ty co nd it ion do es »o r h o ld i mplie s t hat the memb rane a nd t he boun dary layer are c omp one nts o f a reg ulatory syste m capable , if needed, o f switc hin g between elec t rone utra l and electr ogenic sta tes . Th is mea ns that , in the mem br an e bou ndary la ye r, a potential gradie nt mig ht ari se in the dir ection pe rpe nd ic ular to t he membrane that wo uld se rve as a so urce o f add it ional e lectromot ive force and , as such, wou ld ac t like the tran sm em brane poten t ial and e nhance the e lectrogeni c inflow o f nece ssa ry subs tances to the ce ll.
Charge red istribu t ion be twee n the stab le sta tes wit h the form ati o n o f a "s tep" ( Fig. 6) c re a tes potential g rad ients para llel to t he mem bran e , affecti ng the spe ed at which io ns mo ve alo ng the mem br ane and thereby acc e lerating their turnove r . Such an e ffect ma y be o f importa nce in sess ile Chara ce lls .

ne utra l stea dy state. If the sys te m is cl o se to the T uring bifurc ati on, pa tte rni ng is obs erved ( Fig . 4 ) . Th e struct u ral e le ments vary in am pli t ude (cf Fig s . 4 and I) t o th e exte nt depe nd ing on the ini tial perturbation . Th e latter a ppe ars t o mod ulate the a mplit udes o f st ruc tu ral e le ments. The refor e , the re sultin g d istributio ns of ion concentra tions and potent ials in the systern are pert urb atio n-d epe nd e nt (forced ). G iven much ti me, the struct ure s slowly grow , then start to mo ve, afte r which t he sys te m's dynami cs bec omes irreg ular. For ot her pa ra mete r val ues, struct ures are formed t hat do not d epend (o r dep end o nly slig htl y) o n the pertu rba tion ch ara cterist ics a nd pe rs ist for a lo ng lime . The re is a param e ter range whe re, de spite its pro ximit y to the bifurcati o n point , n o structu res aris e i n respon se to a pe rturba tion; the pertu rbat ion is dam ped , but so me times not co m ple tely, le a ving one or more (depe nding o n the pert urbation type ) sma ll seg me nts (F ig. 5 ) w he re ape riodic osc illatio ns of ion co nce ntra tio ns are o bse rved (cf. Fig s. 5 and 2). In yet an ot her paramete r range, a pertur bation app lie d to t he sys tem in the spatiall y ho mo gene ous stead y stat e ca uses t he red istrib utio n of io ns bet ween two stable sta tes in the form of a "ste p," In the state co rrespo nding to the stable foc us of t he mode l w itho ut spa ti al te rms , a wave tra in ca n a rise t hat mo ves to t he step edge .
DI SCUS SION

T he form at io n of small a reas where the system ex hi bits aperi od ic d ynam ic s (Fi g. 5) upset s the charge ba la nce on ly loc all y; beyo nd t hese area s , e lec trone ut ralit y is reta ined . Suc h areas might be a k ind o f ma rker fo r ce rt ai n ion spec ie s, i ndica ting where a reac tio n is to take place . Ampl i tude modu latio n of the str uct ures arising in respo nse to pe rturbati o n o f the hom o ge neo us state nea r the Turi ng bif urcat ion (F ig. 6 ) migh t be o f info rmation sig ni ficance for the sys te m, bec au se thi s effect dep e nds o n the c haracte ris t ics o f the ex te rna l st im ulus ( pe rt u rba ti o n) or, more spec i fica lly, o n the do minati ng frequ enc ie s i n its frequ en cy spectrum. Thu s, co mbina t ion of a no nli nea r c he mical reactio n wit h e lectrodif f usio n proce sse s provid es additio nal en er gy sou rce s in the form of electr ochemi cal grad ie nts for ce ll ion tran sport and produce s a varie ty o f d y na mic mod e s.
R EFERENCES
I. Luca s , W.J. a nd Nuccitclli , R .· Pla n ta , 19 80 , vel. ISO, pp . 120- 13 1.

2. Fis ah n , J . and Lucas , W .J ., Plante , 19 92 , vo t. 186, pr·241-2 4 8. 3. Fisah n, J . a nd L ucas, W.J ., J. M emb rane Bia l . 19 95 . \ '01. I_ . pp . 275- 281. n

The mode l tha t we a nalyze doe s not pu rport to de scribe a real si tua tion ; t he goal of o ur anal ysi s is to sho w that incorporatio n of a se lf-c o ns iste nt fie ld into t he mod el allo ws ne w d y na mic mod es and ne w properties o f the sy ste m to be de tected .

4. Pisahn. J. and Lucas, WJ ., Proc. Na tl. A cad . Sci. USA , 1992, vo l. 89, pp . 3261-3 265.
BIOPHYS ICS Vol. 47 _No.2 2002


REACT ION -ELECTRODlFFU S ION S YS T EM 5 . Tak a , K., H ay as h i , K.· Yo s hida , T. . F uji yo s bi , T .. an d Yam afuji , K.· Eur . Bi 0l'h y s. i .. 1988. vel. 19. pp . 1l -2 1.

271

13 . Loba nov , A J ., Pl yu snina, T .Yu. , R i znic bc nk o . G .Yu .. S tarc zhi lov a. T .K .. and Ru bin , A.B. , Biofiti ka . 2000 , vel. 45, no . 3 . pp . 4 9 5- 501 . 14. Ru b in, A .B ., Biofizika ( Bio phys ic s ), M o sco w: K ni zhn yi Da m U nivcrsite t, 2000. ve l. 2 .

6. Ko ty k. A . and J a nac e k , K .· M embrane Tran sport ; An t nterd isc ip ii na ry Approa ch . New Yo rk : Ple num, 1977 . Trans la ted u nder the Litl e M em branll yi transp o rt. Mo sco w: Mir , 19 80 .

15. O n oleve. P. an d Sc hm idt , S .· O sc illa tio ns a nd Tra "d ing Wa ves in Ch em ica l B e rger . M ., Ed s ., New Yo rk.: u nde r t he tit le Kol ebaniya s is tema kh ; Mo sco w : M ir, 19 8 Sys tems , F iel d. R. and Wiley. 19 85 . T ran sla ted i vo lny v kh imic hesk iJ:h 8. pp . 400-407 _

7. Hod gk in. A .L. and Hu xley , A .F.. J . Ph y s iol . . 1952. vo l. 117 . pp. 500-544 .
8 . La vro ve, A.I. . Plyu snin a, T .Yu .· and Riznichenk o. a .Y u., Mat emat i ka . Ko mput er. Ob raz o vani e , 200 1. vo l. 2 , no . 8. pp . 564-5 72 . 9. Ma lc how . H .· Z,eirs ch rift fUr ph ys ikalis che Chemie , 1998 , vo l. 204, pp . 95- 107 . 10. S e vcl ko va . H., Ko sek. J .· and M ar ek. M .· J . Ph y s, Ch em .· 199 6, vol. 100, pp . 1666- 167 5. 1 1. M un s ter. A .F., H es al , P .· S n i ta, D .· and M arek , M ., Ph y s ical Re view E. 199 4. vel . SO, no . 1, pp . 546--550 . 12 . Mu nster. A .F .. wanal, M ., and S chn eid er . F.W ., Ph y sica S crip ta. 1996 . vel . 67. pp . 5 8--62.

16 . Baren b latt , G .1.. Po do b ie, avto modei 'nos t ', ; prome zhu roc hna )'a a s imptotika (S i m i l ari ty . Sel f -Similari ty , an d Inte rme d iat e As ymptotics ). Len i n grad : Gidrcmeteoizdat . 1982. 2 nd ed .

17. v olose vlcb . P.P. and Levanov , E.l.. A vto mode l ' nye
res h en iya zadacn g azo vo i d inamiki i tep lop er enosa (S el f-s im ilar So lu tions t o P ro blems of G as Dyn am ic s and H eat T ra n sfer ). Moscow : MFfI , 19 97 .
18 . Sam ar sk y . A. A ., Ga la kti o nov . V .A .· Kur d yu mo v, S. P ., and Mikh a ilc v, A.P.· Rezh imy s o b os rren iem v zada cha k h d /ya k vazi/m ein ykh pa ra bo li ch es ki kh ura vn enii ( Ble w- Up Modes i n Pro b lem s fo r Q ua s i-L in ear P ara bol ic Eq ua tio ns) , Mo scow : Na uk a . 19 87 .

BIOPHYSICS

Vol. 4 7

No. 2

2002

...i