J. theor. Biol. (1975) 48, 393-401

Relationship Between the Lifetime of an Enzyme-Substrate Complex and the Properties of the Molecular Environment B. SOMOGYI AND S. DAMJANOVICH Department

of Biophysics, Medical University,

Debrecen, Hungary

(Received 24 April 1974) A theoretical discussion of the decomposition rake constants of enzymesubstrate complexes is presented, based upon an enzyme model published earlier (Damjanovich & Somogyi, 1973). These rate constants are expressed by the aid of molecular parameters characteristic for the enzyme-substrate complexes and the molecules in the surrounding liquid phase. Both the exponential and pre-exponential factors of the expressions describing the composition rate constants contain parameters depending on the mass distribution of the reaction mixture in a specific way which is characteristic for the enzyme-substrate complex. The findings suggest a new kind of the enzyme regulation generated by the surrounding medium. 1. Introduction Earlier papers of the authors discussed the “energy funnel” model of the enzyme action (Somogyi & Damjanovich, 1971; Damjanovich & Somogyi, 1973). According to the model, the activation energy of an enzyme-substrate complex (ES-complex) originates from collisions with the surrounding molecules and the energy uptake takes places at the surface of the complex. These collisions must occur with sufficient energies and at particular sites of the surface of an ES-complex at appropriate times to cause activation.The model supposes a coupled oscillator system, depending on the structure of the enzymic protein, which conveys some part of the energy to the active centre. The present paper considers a possible relationship between the mass distribution of solvent molecules and the decomposition rate constants of ES-complexes. The following conditions are assumed in order to decrease mathematical difficulties : (1) Dissociation of the ES-complex either to enzyme + substrate (E + S) or enzyme+product (E-t Pr) demands excitation at particular sites of the complex. A colliding particle-exciting the complex at particular sitesconveys translational energy to the ES-complex. The kinetic energy of the 393

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particles is likely to be transformed into vibrational energy of the complex, or vice versa (Stretton, 1969). Any other kind of energy uptake will be included in a probability factor, indicative of the efficacy of collisions. (2) The particular excitation sites and energy transfer routes of the complex are the same for ES + E + S, and ES + E f Pr transitions. (3) t1 = 0 is the time of the first collision in the sequence which brings about excitation. All of the further collisions causing excitation will occur at times t2, t,, . . . , tk. It is assumed that t, < t, < . . . < ty where k’ is the number of excitation sites for a complex. (4) The colliding particles must exceed a velocity threshold to cause no more than a translational-vibrational transition (Stretton, 1969). This is why the ith exciting site of the complex can be excited only by a threshold velocity vi for the ES --) E+S and wi for both ES -+ E+S and the ES + E+Pr transitions. vi Qwi are the velocities of the colliding particle suitable for the ith particular excitation site. (5) The solvent, i.e. the liquid environment, is modellized by a rectangular three-dimensional lattice (Somogyi, 1971). Solvent particles vibrate and rotate in each lattice point for z average time and reach the next lattice point by a “jump” in a negligible time compared to r. If 1 denotes the distance of two nearest lattice points

where D is diffusion constant of a given solvent particle (Somogyi, 1971). (6) A series of suitable solvent molecules-or particles-will be ordered to each particular excitation site of the ES-complex with radius ip1 < Pj < ipz. Here thej = 1, 2, . , . , Ji, where Ji is the number of the types of solvent molecules capable of acting at the ith exciting site of the ES-complex. The limits of radius ipl and ipz are characteristic for the ith exciting site of the complex. It can be accepted that only one lattice point fits this site as the starting point for each molecule which causes excitation. (7) Considering a very simple reaction scheme: E+&

the individual

ES:E+Pr k-1 rate constants have the values: k2 = $ k-,

=y

(2) (3)

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where t’ is average lifetime of the ES-complex and L is the probability of the ES -+ ES Pr transition, in the case of decomposition of the complex (Somogyi & Damjanovich, 1971). 2. The Model The average dissociation rate of an ES-complex 1 1 2=t,

- P,.

(4)

Here t, is the time between the beginning of two successive and suitable collision patterns which can cause activation. Ph shows the probability of excitation if the suitable lattice points contain appropriate molecules. Since individual collisions generally do not occur simultaneously, a time transformation makes mathematical complexities simpler. Let tt = t-(ti-tl), i = 1, 2, 3, . . . , k’ (5) where t, is transformed time, t is “ordinary” time and ti is the time of collision at the ith site. By this transformation the collisions can be treated as if they occurred simultaneously at the time of the first collision in the volley which results in an efficient activation. Let r, be the average time interval in the transformed time during which all of the suitable lattice points contain one appropriate molecule and let 8, be the average time lapse between two such successive intervals. If P, is the probability of occurrence of an activating collision pattern. pt -=-

l-P, 4

Tt

*

(6)

It is clear that t, = ZtSflt,

(7)

or from equations (6) and (7)

If the solvent particles in the available space follow a Poisson and if there is a finite number of types of these molecules on their radii (masses), then the probability of occupation of the ith by an appropriate molecule-satisfying condition (6)-can be Pi =

distribution, the basis of lattice point written :

C CjVj emcjvJ. j=l

Here Ji denotes the number of types of appropriate molecules and Cj and Vj are concentration and free volume of a jth type molecule (Somogyi &

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1971). Since for every type of molecule CjVj ~ 1 (10) of occurrence of an activating collision pattern [see equation P, = fi i=l

% cjvj.

(11)

j=l

The accidental interaction of near-by colliding particIes is not taken into account. To calculate rt, the mean residence time in which all of the suitable lattice points for the specific excitation sites of the ES-complex contain one appropriate molecule, the exponential distribution of the probability of emergence from one lattice point is considered (Somogyi, 1971). Thus, if suitable particles are found at the correct lattice points at the transformed time t,, the expression: 1 - exp

-Ati 1 d@il zi (12) ( i=l’ > gives the probability that none of the particles emerge from the lattice points in time t,+At and the particle at the ith lattice point leaves it in the time interval (t,+At, t,+At +d(At)). zi is a mean residence time of appropriate molecules [as defined in condition (6)] at the ith lattice point, ordered to the ith excitation site of the ES-complex. The quantity may be written: zi

(13) where zj is the mean residence time of a jth type particle at a lattice point. Using equation (12), 7t can be calculated by summing for i, and averaging for time:

xt = iil Integrating

From

t $ (At> exp (--(At> t

i=l

i) 44.

(14)

the equation (14)

equations

(1) and (13) and using the Einstein-Stokes

relation

D = kT/6qp: ni1’rai zi = kT

(16)

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in which

(17) i.e. it is the weighted average of molecular radii belonging to those molecules which are suitable for the ith excitation site. The time, t,, of equation (8), between the beginning of two relevant collision patterns which activate the ES-complex, can be derived using equations (ll), (15) and (17):

TV =g( jjl ii)-’ (A jlcjYi)-l* Average dissociation rate of an ES-complex may be found as follows. First the efficacy of the collisions (P,J must be determined. The quantity may be divided into two parts:

P, = P, fi eP+ i=l

(19)

P, expresses the probability

of the sterically satisfactory collisions and also contains the probability that it occurs at the required time. It includes also all of the probabilities for different energy absorptions mentioned in condition (1). ,Pi defines the probability of occurrence of an energetically suitable molecule, to excite the complex for transition ES + E+ S, at the lattice point for the ith excitation site of the ES-complex. .Pi is obtained by averaging according to condition (4):

p. = j$l cjVj e-(mJZkT)Y*’ eI jitl civj

(20-l

Using equations (18)-(20), equation (4) becomes : 1 F= Ps-$(iil

JJ,fJ I 1

$ CjVj e-(mj'2kT)V'Z. (21) j=l of the constants k2 and k-, demands a detailed

The determination evaluation of L, the probability that an ES-complex becomes E+Pr. Let e be the dissociation probability of an activated complex which possesses the activation energy for the process ES + E +Pr, and let P be the probability that an ES-complex, being activated for a dissociation into E+ S, has the activation energy also for the transition ES + E+Pr.

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According to these L = p. The conditional represented by

probability

p=

fij=l i=l

(22)

P, according to conditions

(2) and (4), is

2 cjVj exp ( - $$,(w; -$)) ____-.

(23)

Thus, rate constants k2 and k- 1 can be described by using equations (21)-(23) as well as (2) and (3) :

[ilCjQeXP(

-~(“i-vi))][~~Cj~eXP(

-$$)]

xfi.

i=l

%

Cj~

j=l

g x

(24)

CjVj

eXp

( -

2~(Wf-V~))

l-efi.i=l

(25) i=l j$l

i

For the sake of simplicity

cjvj

I

’

we introduce the symbols Ed, EP and p as Ji

,-MT

_ -

ikfj~lcjFe~~(p~+v~)

(26) jlfl

'.ivj

)

iI

cjvj

ev

(

-

,f&d)

(27)

-P1 =( iil i > iij j$l‘jvj

(28)

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It is necessary to note that while the values Ed and J!$ are actually obtained by averaging, this method cannot be used to obtain jj. A limit value for fi can be found when only one type of molecule is in the solution. jj = p/k’, where p is the radius of the solvent whilst k’ gives the number of the excitation sites of the ES-complex. Hence it is obvious that p denotes a parameter depending on the mass-distribution in the solution and on the characteristics of the ES-complex also. It has been shown so far that the interpretation of parameters Ep and Ed differs considerably from that of the earlier models (Eyring & Eyring, 1963; North, 1964). Because of the progressive accumulation of the energies E, and & these components of the total energy required may not be realized at the same time (some energy may be dissipated before further absorption can occur). Using the above symbols equations (24) and (25) may be put into more concise form : kT - E,IkT k2 = P,