Statistical mechanics of antibody-antigen binding: affinity analysis

PHYSICA ELSEVIER

Physica A 218 (1995) 214-228

Statistical mechanics of antibody-antigen binding: affinity analysis * R a j a n i R. Joshi Department of Mathematics and School of Biomedical Engineering, Indian Institute of Technology, Powai, Bombay, 400 076 India Received 1 February 1995

Abstract

A model of the statistical mechanics of antibody-antigen binding is presented. A relation between the mutual affinity of these molecules and the most probable state of their bound complexes is derived. Analysis of the involved nonlinear optimization problem is presented along with some applied computational results. Importance of the approach for other applications is highlighted.

1. Introduction

Statistical mechanical analysis of biomolecular interactions is important because of its applications in understanding the energy based structure-function relations in various biophysical/biochemical processes. In order to correctly predict and explain the binding properties of molecules in a given environment the partition function for corresponding system ought to be known for the Statistical Mechanical analysis. This is a mathematically intractable problem for the biomolecules because of the aperiodic and macro structures of the latter. The structural complexity of the biomolecules together with their functional dependence on the surrounding medium further limits the applicability of the classical models and theories in statistical mechanics. These difficulties pose challenging research problems in theoretical biophysics. Recent work of Wang (1990) on the statistical mechanics of protein-ligand binding had offered some important applications in enzymology. Using this approach we had * Partly supported by a project-grant from the Department of Biotechnology, Government of India. 0378-4371/95/$09.50 (~ 1995 Elsevier Science B.V. All rights reserved SSDI 0378-4371 ( 9 5 ) 0 0 0 8 8 - 7

R.R. Joshi/Physica A 218 (1995) 214-228

215

studied (in Dar (1993)) the antigen-hapten binding processes for data available from real experiments. In this paper we have considered the statistical mechanics of antibody (Ab) and antigen (Ag) binding. We have derived a mathematical relation between the mutual affinity of involved molecules and the most probable state of their bound complexes. We have analysed the associated feasibility and optimization problems and presented applied results obtained thereby. Mathematical formulations for the model system under consideration are given in Section 2 below. The relation between Ab-Ag affinity and the most probable state is analyzed in Section 3. Applied computational results are presented in Section 4. Importance of the present work in molecular immunology and its scope in other applications is discussed in Section 5.

2. Optimal states of Ab-Ag complexes In a given system of B free Ab's and G free Ag's, a random number of Ab and Ag molecules may bind to each other non-covalently forming an aggregate, called an ( i , j ) m e r of i Ab molecules bound to j Ag molecules; i = 0, 1. . . . B; j = 0, 1. . . . G. Using multi-type branching processes Macken and Perelson (1985) had derived the expression for the probability distribution of such (i, j)mers as a function of the A b - A g binding coefficient. Here we formulate a statistical mechanical model to obtain the probability of attainment of states accessible to these ( i , j ) m e r s in a given environment. 2.1. Description of states The difference in an ( i , j ) mer and a single macromolecule arises in the strength of bonds that hold different constituents together. In the formation of a macromolecule the combining molecules lose their identity due to the extensive overlap of their electron clouds. It therefore is not appropriate to describe the (quantum) state of a macromolecule by independently specifying the states of its constituent atoms/molecules. However, in an ( i , j ) mer, the antibodies and antigens bind to each other by weak non-covalent bonds formed mainly due to Van Der Waals forces and electrostatic forces between atoms present at the binding sites (e.g. in Volkenstein (1983), Hoebek et al. (1987)). Here the antibodies and antigens themselves maintain their 'identity' in the sense that there is no overlap of their electron clouds. Hence the states of an ( i , j ) m e r might reasonably be described in terms of the states of its constituent molecules. The state of an (i, j ) m e r will be described in our models by specifying the states of each of the i antibodies and j antigens forming this aggregate. In order to include the feasibility of all possibile mechanisms of aggregation, the numbers of bound sites would be considered as random variables and the (quantum) states of the involved molecules would be identified in terms of these random variables.

216

R.R. Joshi/Physica A 218 (1995) 214-228

2.2, Mathematical formulation To begin with, in model I we consider the simplest situation where all the sites of an Ag molecule (respectively of an Ab molecule) are identical. The system consists of G g-valent Ag's and B f-valent monoclonal Ab's specific to these Ag's. The surrounding medium is in equilibrium and has no binding interactions with the constituent molecules. A more complicated situation where the binding sites of the Ab's (and/or the Ag's) are not identical is formulated in model II.

2.2.1. Model I: non-distinguishable binding sites Consider B f-valent Ab's and G g-valent Ag's in an aqueous medium at a constant temperature and volume: due to mutual affinity based interactions and binding of the Ab and Ag molecules, this system, at equilibrium, would consist of different (random) numbers of ( i , j ) m e r s where, i = 0, 1 . . . . B and j = 0, 1,2 . . . . G; noting that i and j both will not be zero simultaneously and that a free Ab would be considered as a (1,0) mer and a free Ag as a (0,1)mer. As stated above, the state of an ( i , j ) mer would depend on the quantum states of its constituent Ab's and Ag's. Hence a state, say k of an ( i , j ) m e r would be defined by:

k = ( q s ( a s ) , s = 1. . . . i

and

Um(t~m),m =

1. . . . j)

===> (sth Ab is in quantum state qs and has as sites bound,

s =

1

.... i

and mth Ag is in quantum state Um and has tim sites bound, m = 1. . . . j ) . These allowed energy eigenstates of the Ab's and Ag's are considered to be well defined in terms of the quantum mechanics of the individual molecules. Let Eq,(a,) and Eu,(#.) be the energy eigenvalues associated with the quantum states qs(as) and u.,(flm) of the Ab and Ag molecules respectively. Let the degeneracies of these eigenvalues be nEq, (as) a n d n~um (]~m) respectively. Then the total number of complexions for the ( i , j ) mer with sth Ab and mth Ag in energy levels Eqs(as) and Eu.,(#.,), respectively, would be given by: i

j

J'~k(i'j) " - - - ' H H [nEq ~ (as)nte . ( ~ m ) ]i[j[. s=-I m=l

(1)

The factors i! and j! arise because any of the i Ab's and any of the j Ag's could be considered as the sth Ab and mth Ag (denoted by Abs and Agm), respectively. The total number of states available to the ( i , j ) mer is given by: i k

1

(as) (tim) Eqs(as) Eum(flm) s=l m=l

where ( a s ) denotes different feasible values of the vector (cq, a2 . . . . . ¢Xi) etc. Summation over Eu.,(flm) and Eq.(as) takes into account all possible energy values that can be assumed by the Ab's and the Ag's with as and tim sites bound respectively.

R.R. Joshi/Physica A 218 (1995) 214-228

217

Hence the probability of occurrence of an ( i , j ) m e r in energy level k is B

pk(i,j)

(~(i,j) / F). ~k /a~tot,

G

. ~ ~ ()(i,j) -otol = "~tot • i--O j=O

(3)

2.3. Optimal (most probable) state It is important to note here that, -o~i,j) being linear in the degeneracies of energy levels will be maximum for those a'ss and /3~,,s which identify energy states with maximum degeneracy. Considering that all quantum states qs(.) and q~(.), etc. would a priori be equally likely to be realized by any Abs (similarly Um(.),Utm(.), etc. would be equally likely for any Agm), we would let the state type q and u as dummy indices in the analysis to follow. Hence the optimisation problem would mainly be focussed at finding the optimal solutions for as, ~,,; s = 1. . . . . i, and m = 1 . . . . . j for the given system. Further, by definition, the as and/3,, would be subject to certain constraints in order to ensure the formation of an (i,j)mer. Maximisation of .Ok must be carried out subject to these constraints.

2.3.1. Binding constraints 1 <_a~,
l<_/3,.
(4)

as, /3m are integers; s = 1 . . . . . i; m = 1 . . . . . j. This constraint ensures that the total number of bound sites as (or/3m) of any Ab (or Ag) is not larger than the maximum number of binding sites available on an Ab (or Ag). At least one site of an Ab (or Ag) must be bound for it to be a part of an ( i , j ) m e r ; the lower limit on a,. and /3,. ensures this, i

j

Zas=E/3m s=l

(5)

m=l

The above constraint ensures that the ( i , j ) mer would not expand to ( i t , j ' ) m e r for it > i a n d / o r j r >j, S

M

S + M < ~--~as+ ~ f l m S=1

<_Sf + Mg,

(6)

m=1

for S = 1 . . . . . i - 1 and M = 1 . . . . . j - 1. The violation of the above constraint would lead to the formation of an (S, M ) m e r which will not have any room for further A b - A g binding for possible formation of an (i, j)mer. It may be noted that the solution of (411 would also satisfy (6).

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R.R. Joshi/Physica A 218 (1995) 214-228

2.4. Model H: distinguishable sites As a generalization o f model I we consider here a more realistic situation, where each binding site of an Ab or Ag molecule would have an identity (due to the specific radical attached to this site, for instance) called a 'type' which distinguishes it from other sites on the same molecule. Let there be h possible 'types' of the f-sites on the Ab molecule and c possible 'types' o f the g-sites on the Ag molecule under consideration. The notation qs(as), urn(fin) used in the model I would then be modified as qs((V)s,m) and U m ( ( V ) s , m ) where s = 1 . . . . . i and m = 1 . . . . . j where (V)s,m is an h × c matrix whose (t,t~)th element say (V(t't'))s.m denotes the number of 'bounds' where a site of type t on the Abs binds to a site of type t' on the Ag m. It may be noted that a~ (respectively fin,) of model I would be interpreted here as

aS =

( v ( t ' t t ) )s,m

(7a)

,

m=l \ t=l F=I

~., =

Z ( V (''t') ) s,m • s=l

(7b)

t=! F=I

The role o f as,/?m; s = 1 . . . . . i and m = 1 . . . . . j as decision variables for the involved optimisation problem in model I would now be played by the variables (V(t't'))s,m; t = l . . . . . h a n d t ~ = 1 . . . . . c ; w i t h s = l . . . . . i a n d m = l . . . . . j. The situation where specific 'type' say t* of an Ab site cannot match with a 'noncomplement' type say t*' of an Ag-s!te would be a particular case of above model namely, the decision variables models.

(V (t',t"))s,m will be explicitly set equal to zero in such

3. Affinity and optimal state analysis By definition, the probability of the optimal (equilibrium) state, for any (i, j ) m e r in model I would satisfy f'2opt i'j) /~Wtot /1")* = ('Oij

where

(8)

toij (from Macken and Perelson ( 1 9 8 5 ) ) is the equilibrium probability of an

(i,j) mer as given below: B

toiO=B+G(1-rp)g, G

wOJ=B+G(1-p)

f,

i=1; j=l;

wi0=0,

i>2,

(9a)

w0j=0,

j>2,

(9b)

R.R. Joshi/Physica A 218 (1995) 214-228 wij =

219

f G r J - l ( i + J) t (g-l)i,~ ~fl (f-I)j-i+l (B+G)j ~j-1 , ( - l ) j ) P i + j - l ( l - p ) × (1 - r p ) (g-1)i-j+l,

i > 1,

j _> 1,

(9c)

where p = Pr (a randomly chosen antigen site is bound), and r is the ratio of antigenic sites to antibody sites (= f G / g B ) . Here ,~opt°~i'J)corresponds to the expression for s2~i'j) (Eq. (1)) for the optimal state k with the (as) and (/3m) involved obtained by solving the associated feasibility problem; S21*,t is as defined in Eq. (3).

3.1. Estimation of O~, Noting that a (0,1)mer represents a free Ag, we can estimate S2~ot as equal to (using Eq. (8) above ) S2Ag/~O01 where w01 would be as given in Eq. (9b) above and OAg would be equal to the n t value (known from the experimental data) associated with the equilibrium energy of a free molecule of the antigen under study.

3.2. Estimqation of O,,pt Using the above estimate of S2tot in the equilibrium relation (8), the problem of finding S2,,pt reduces to the following: For given B, G, f, g, p, ~- find integers i*, j*, such that

toi.j. = Maxi,j{~oij [ 1 < i < B, 1 < j < G}. Then ~-2ol't = O')i* j* ~'~t*ot"

( ! O)

The optimal solutions for (as) and (/3,,); s = 1. . . . . i*, m = 1 . . . . . j* involved in the binding constraints would be obtained by solving the corresponding Feasibility Integer Programming Problem represented by the set of constraints (e.g., Eqs. ( 4 ) - ( 5 ) for model I above) on these decision variables. Let {O'}i.b, {/3}j*b denote the bth distinct set of these feasible solutions for optimal (i*, j*).

3.3. Ab-Ag affinity and optimal bound state Using the arguments given in section 2.3 and noting that for a given configuration

{i,j, {~}i, {/3}j}, the roles of (i) the potential energy (in terms of structural match between an Ab and an Ag) would be governed by the affinity parameters (namely, the binding probability p in our models) and (ii) the internal energy would be represented by the binding reaction rates (say O) at equilibrium in a given medium at a given temperature, we would consider nE,~,(as) and n'Eu m (tim) in Eq. (1) as functions (say ~ ( . ) and qt(.) respectively) of as (or, tim), P and 0. Then we would have

220

R.R. Joshi/Physica A 218 (1995) 214-228 j~o(P' i,J )

i j = H H q~(as,p, O)~(flm,p, O)i!j!.

(11)

s=l m=l

The appropriate maps of the involved energy-degeneracy functions can now be computed by applying multiplex polynomial fitting on q~(.) and !/' (.) using different sets of parameters B, G, f, g, r, p and /2As and corresponding optimal solutions I'2opt (obtained as a function of {i*,j*, {_a}/.., {fl_}j..} from Eq. (10) above).

4. Computations We have carried out several computational experiments on model I to illustrate the applicability of the above analysis. For moderately large values of B and G the maximisation of w~ (= o~ij x ,O~'ot) with respect to i,j;i = 1. . . . . B and j = 1. . . . . G was carried out using Stirling's formula for factorials in the approximation based Branch and Bound formulations. The IMSL code for the nonlinear programming algorithm NLPQL of Schittkowski (1986) was used for the necessary computations. For sufficiently large values of i, j a Poisson approximation of the binomial terms was taken in the expression for w~. A direct search monotonicity analysis (mathematical) for the local maximisation (along the lines i = j × constant) of wq gave us the following interesting results which made the computations of i*, j* much easier:

Lemma 1. Let C1

e rp(g-l) p ( f - 1)

C~

1 ~11'

e p(f-l) C2 = ~.p(g_ 1)'

C~

1 C2

Then (i) for C~ _< 1 and C2 _> 1, to~ is a monotone decreasing function of i, j if i/j E

[Q*,CE]; (ii) w~. is increasing with respect to j and decreasing with respect to i on the set {(i, J) I i/j > C2}; (iii) w~ is increasing with respect to i and decreasing with respect to j on the set {(i, J) I i / j < C~}. and hence,

Lemma 2. With Cl and 6"2 etc. as defined above and denoting by [y] the integer nearest y V real y, we have the optimal solutions (i*, j*) satisfying the following: (i*,j*) E F,

wi.,j. = M a x ( i , j ) E r { W i , j } ,

F = {( [B*], [G*] ), ( [Q*], [S*] )} U { ( i 0 , j 0 ) I (io,jo) defined below}. (i)

B*=Min1
R.R. Joshi/Physica A 218 (1995) 214-228 Table 1 Optimal solutions for different sets of parameters. For each set, computational experiments.

~(~AGw a s

221 set equal to 1.5 × 1025 in the

Parameters

Optimal solutions

Set number

p

f

g

B

G

I 2 3 4 5 6 7 8 9

0.025 0.155 0.515 0.035 0.065 0.615 0.700 0.015 0.015

5 5 5 5 5 5 5 7 10

2 2 2 2 2 2 2 2 2

175 175 175 175 175 175 175 750 750

100 100 100 100 100 100 100 100 100

i* 9 9 10 13 13 13 16 8 13

j* 10 18 16 7 12 13 10 11 7

~lopt = wi*,j* 12~ot 2.75 ×109 1.66 ×1012 7.1 ×1018 3.08 ×1011 6.72 ×1013 3.8 × 1022 3.2 ×1024 1.68 xl0 l 1.85 ×109

(ii)

Q* =MaXl
io = [ H ( x * ) x * ] ,

jo = [ H ( x * ) ] ,

where x* and H ( x * ) are such that

Maxx{o~H
=

¢-O~H(x.)x*][H(x*)],

1 <_jx
4.1. Applied results The optimal solutions for i*, j* for different sets o f parameters (from the data in Joshi (1992) ) are shown in Table 1. For each such solution pair, the optimal solutions { a } i . , and {_0ff}J* o f the c~ and t h e / 3 were obtained by solving the feasibility IPP given in ( 4 ) - ( 5 ) with i replaced by i* and j by j*. An integer partition subroutine developed by us (Joshi, 1990) was used for this purpose. The optimal solutions computed thereby are given in Table 2. All the computational experiments including the multiplex fitting of the functions q~(.) and ~ ( . ) were carried out on a C D C C Y B E R 180/840 machine at IIT Bombay. The variation in the S2opt value with respect to the affinity parameter p is illustrated in Figs. 1 and 2 for different A b - A g complexes. The estimated functional forms o f 4 ( . ) and ~ ( . ) are shown in Figs. 3 and 4, respectively.

R.R. Joshi/Physica A 218 (1995) 214-228

222

Table 2 Total number of solutions ({a}i*, {/3}j.) of the IPP ( 4 ) - ( 5 ) for different sets of parameters are shown along with some samples of these solutions. The notation used here should be interpreted as follows: {¢r}i = { 1 x k, 2 x j } means k of the as are each equal to 1 and j are equal to 2; k + j = i; s = 1,2 . . . . . i etc. Set number Total number Some of the solutions ({a}i*, {,8}j. ) (as in Table 1) of solutions (From the total number of solutions obtained for a parameter set, ( { a } i . , {,8}j. ) at most six randomly sampled solutions are given below.) 1

35

({1 .... .... ({2 ....

2

Ol

({2x9},{1

3

13

({l ({1 ({1 ....

4

169

({1 x 13},{1 x 4 , 2 x 1 , 3 x 1 , 4 x 1}), . . . . ({l x 13},{1 x 5,3 x 1,5 x 1}), . . . . ({1 x 10,2 x 3 } , { 1 x 3,3 x 3,4 x 1}), . . . . ({1 x 10,2 x 3},{1 x 4,2 x 1,5 x 2}), . . . . ({1 x 6 , 2 x 7},{1 x 3 , 4 x 3,5 x 1}), ...({2x 13},{2x 1,4x6}) .....

5

201

({1 x 13},{1 x 1 1 , 2 x 1 } ) . . . ( { 1 x 9 , 2 x 4 } , { l . . . . ({1 x 5,2 x 8},{1 x 7 , 2 x 3 , 4 x 2 } ) , . . . .

x 8,2 x 1},{1 x 10}) . . . . ({1 x 6,2 x 3 } , { 1 ({1 x 5,2 x 4 } , { 1 x 7 , 2 x 3 } ) , ({1 x 5,2 x 4 } , { l x 8 , 2 x 1 , 3 x 1}), x 9},{1 x 6,2 x 1,3 X 2,4 x 1}), ({2 x 9},{1 x 8,5 x 2}) . . . .

x 9,3 x 1}),

x 18})

x 4,2 x 6},{1 x 16}),({1 x 3,2 x 7},{1 x 15,2 x 1}), x 2 , 2 x 8},{1 x 1 4 , 2 x 2 } ) , ( { 1 x 2 , 2 x 8 } , { 1 x 15,3 x 1}), X 1,2 x 9 } , { 1 x 15,4X 1}), ( { 2 x 10},{1 x 1 3 , 2 x 2 , 3 x 1}) . . . .

x9,2x

1,3x2}),

({l x 5,2 xS},{l x 5 , 2 x 5 , 3 x 2 } ) ,

. . . . ({1 x 1,2 x 12},{1 x 7,3 x 3,4 x 1,5 x 1}), . . . . ( { 2 x 1 3 } , { 2 x 1 1 , 4 x 1}) . . . . 6

172

({1 .... .... .... ....

x 13},{1 x 13}),({1 x 12,2 x l } , { l x 12,2 x 1}), ({1 x 9,2 x 4},{1 x 12,5 x 1}), ({I x 5 , 2 x 8 } , { 1 x 11,5 x 2 } ) , ({1 x 5,2 x 8},{1 x 8,2 x 2,3 x 3}), ({2 x 13},{1 x 7,2 x 2,3 x 2,4 x 1,5 x 1}) . . . .

7

347

({1 .... .... .... .... ({2

x 16,1 x 5},{2 x 4,3 x 1}), ({1 x 14,2 x 2},{1 x 7,2 x 1,4 x 1,5 x 1}), ({1 x 9,2 x 7},{1 x 1,2 x 7,3 x 1,5 x 1}), ({1 x 5,2 x 11},{1 x 4 , 2 x 1,3 x 2 , 5 x 3 } ) , ({1 x 3,2 × 13},{1 × 4 , 2 x 1,3 x 1 , 5 x 4 } ) . . . . . x 16},{2 x 4,3 x 2,4 x 2,5 x 2 } ) . . .

8

19

({1 ({1 .... ....

x 5,2 x 3},{1 x 11}),({! x 4,2 x 4},{1 x 10,2 x 1}), × 3,2 x 5},{1 x 10,3 × 1}), ({1 x 3,2 × 5 , } , { I x 9,2 x 2}), ({1 x 1 , 2 x 7 } , { l x 8 , 2 x 2 , 3 x 1}), ({2x8},{1 x9,2x 1,5x 1})...

9

572

({1 x 13},{1 x 5 , 2 x 1 , 6 x 1}), . . . . ({1 x 1 1 , 2 x 2 } , { 1 x 5 , 2 x 1,1 x 8 } ) , . . . . ({! x 11,2 x 2},{2 x 6,3 x 1}), . . . . ({1 x 8 , 2 x 5 } , { l x 4 , 2 x 2 , 1 0 x 1}), . . . ( { 1 x 7 , 2 x 6 } , { l x 5 , 4 x 1,10X 1}), . . . . ( { 2 x 13},{1 x 3,3 x 1 , 6 x 1,7 x 2 } ) . . .

R.R. Joshi/Physica A 218 (1995) 214-228

223

400

350

300

T

250

sEI I

200 SET 2 150

I00

50

I

I 0.20

I

I 0"4 0

I

I 0.60

I

I 0.80

I

I 1"0

P

Fig. 1. 12~oip"j*) ~ as function of p for optimal (i*,j*) obtained for different sets of parameters in Table 1. Computations were done using multiplex polynomial forms of q,(.,01) and q~(.,02) in Eq. (11) with 01 = 0.022 and 02 = 0.053. 5. Discussion In this paper we have derived a theoretical relation between the mutual affinity of antigen-antibody and the optimal states of their bound complexes. Applied computational results have shown here that affinity coefficient plays a significant role in governing the optimal size and the binding patterns of the Ab-Ag complexes and hence in the statistical mechanics of the involved molecular interactions. Recent immunological findings indicate the importance of the size and concentration of the bound complexes in the regulation of immune response. Our approach would help quantitative elucidation and hence deeper understanding of the immune-functional control by the molecular structural features (responsible for the specific 'affinity') and the energy constraints. We have also presented a method for estimating the number of complexions (associated with a quantum state) in which an ( i , j ) m e r would optimally reside with a given affinity parameter and a constant binding reaction rate. The associated optimal

R.R. Joshi/Physica A 218 (1995) 214-228

224

450~

400

t~

350

3O0

2

250

._.~ 0

~

200 x 0

150

f

2 1

f

I00

t

50 0

I 0.20

I

I 0.40

i

I 0.60

I

I 0o80

I

J 1.0

la -------p, Fig. 2. o(i'J) --opt as function o f p for different (i,j). Computations were done using Eq. ( 1 1 ) with multiplex polynomial forms o f 4 ( { o r } i , p, Ot ) a n d 1/I ( {/3}i, p, 02) for 81 = 0.022, 02 = 0 . 0 5 3 a n d {t~}i, {/3}j b e i n g a feasible solution for the corresponding p a i r (i,j). C u r v e 1. i = 1, j = 1; c u r v e 2, i = 5, j = 5; c u r v e 3. i = 10, j = 5; c u r v e 4. i = 10, j = 10; c u r v e 5. i = 25, j = 25; curve 6. i = 25, j = 30; c u r v e 7. i = 25, j = 50.

energy function can be computed by using the Eqs. (A.1)-(A.10) stated in Appendix A. Applications on real data in this regard will be reported subsequently. Computational techniques developed here would be of direct utility in studying the statistical mechanics of other macromolecular interactions. Extension of the approach to include the bond lengths, angles of rotations and other structural features along with the number of bound sites would give rise to further theoretical research problems in the applied as well as the conceptual domains.

R.R. Joshi/PhysicaA 218 (1995) 214-228

225

700

God

500

"~400

c~

~300 o

200

I00

i

!

I

2

i

4 OC---,~

!

!

6

I

|

8

Fig. 3. q~(a, p, 01 ) as a function of a for 0! = 0.022. Curve 1. p = 0,01; curve 2. p = 0.1; curve 3. p = 0.3; curve 4. p = 0.5; curve 5. p = 0.7; curve 6. p = 0.9.

Appendix A. Energy computations T h e energy o f an ( i , j ) m e r

c o u l d be expressed as a sum o f its internal energy and

potential energy arising due to the i n v o l v e d binding reactions (details in Dar ( 1 9 9 3 ) ) ,

Eq,(as) =E~°t(Ots) +E~t(as),

E..,(/3m)

= E~'°,t(/3.,)

+E~n'(/3m).

(A.I)

Further, noting that the potential energy level contributed by any pair o f c o m p l e m e n t a r y m o l e c u l e s say Abs and Ag,,,s m u s t ' m a t c h ' for m a k i n g the interaction ( l e a d i n g to b i n d i n g o f their respective as and /3., sites) b e t w e e n t h e m feasible, we w o u l d have:

R.R. Joshi/Physica A 218 (1995) 214-228

226

700

600

I 500

g °~ n

400

x

o 300

200

I00

I

I

2

|

I

4

,

I

6

p---~ Fig. 4. ~ (,8,p, 02) as a function of fl for 612 ---- 0.053. Curve 1. p = 0.01; curve 2. p = 0.1; curve 3. p = 0.3; curve 4. p = 0.5; curve 5. p = 0.7; curve 6. p = 0.9. ~ ( s) ppot ms ~ q s

0<~(s)

( ,a, "~ _ 12( m ~ ) lZ;flot \ ~S ) -- ~S --Urn s '

<1

0<(~m')

< 1;

s = l 2,

,i,

m s = l 2,

j.

(A.2)

r/.~,~) and ~¢~m.) denote the fractions of potential energy of Abs and Agm, respectively, utilized for their binding with Agm' and Abs, respectively. Clearly if Abs binds with Agm2 . . . . .

Agm~ t h e n

~ - ~ _111711, ( s ) = 1, b=l

(A.3)

R.R. Joshi/Physica A 218 (1995) 214-228

227

and if Ag,,, binds with Absl . . . . . Abe.,, then /ii

s¢~.~') = 1.

(A.4)

b=l

Hence summing up Eqs. (A.2) and considering Eqs. (A.3) and (A.4) we would get the following resultant constraint: i

j

~.pot "~ £2, q ~ :I..g S 1 = ~ r P ~u,,°t(t~ ,t-'m,.

~

~ .

s= 1

(A.5)

m= 1

Similar reasoning would lead to the following resultant relation between internal energies: i Eq, ,nt ( oq. )

-

-

j i iEab q- ~-~Eu,,(1~m)_jEag.~(~--~Ols)ffsite, int

s=l

j=l

(A.6)

s=l

where EAb and Egg are the energies of free Ab's and Ag's respectively in their most probable states. Change in internal energy may exist because of other interactions (possibly due to the presence of other molecular species in the system) - not leading to binding effects; esite would correspond to the average energy associated with such effects. A. 1. Total energy constraint Applying above reasoning (of sharing of energies between each bound pair in an ( i , j ) m e r ) to the given system of B antibodies and G antigens we would get the following equilibrium constraint on 'resultant' expected energy:

a=a,

~-~ ~f'~D(i'j)~ ra,r r=rl(a) i=1 j=l

:(O,)Osfq.,(ol~). -~ ~'~(~m)~/mEum(~n,) m=l

= Etot,

(A.7)

where Etot denotes the energy of the system; 0, and T,, are random variables on the real interval (0,1) with the respective probability density functions fi(Os) and fi(Tm); these denote the random fractions of energies shared by a bound pair of Ab, and Abm,

aj = 1,

a2 = Min{Gf, B}

rl(a) =Max{G,[a/f]},

r2(a) = M i n { G , ga},

(A.8)

and, V a = a l , al + 1 . . . . . a2 and r = r l ( a ) , r l ( a ) + 1. . . . . r2(a), i,, ji f i O')ain,rjn n(i,j) ra.r = O')10B--a( O o 1 G - rt zaB)xtzrG x) X ~--'K~-" z_.~z..~ a!r! &Jr !r./,,!' i=1 .j=l n=l ai,,

(A.9)

228

R.R. Joshi/Physica A 218 (1995) 214-228

where ia = number of distinct partitions of a, Ji = number of partitions of r which 'match' the ith partition of a ; ai,, and r j,, denote respectively the nth entry in the ith partition of a and corresponding jth partition of r. The above probability distribution was derived in Chandrika and Joshi (1990); the details about matching partitions of i and j above are given in Chandrika and Joshi (1989). The expressions for to O and other quantities are as given in the text. A.2. Possibility o f medium interaction In case the medium is capable of exchanging energy with the Ab's and the Ag's, the system forming their (i, j)mers would posses energy Etot with probability e -e'~/Kr where T denotes the absolute temperature of the medium and t¢ denotes the Boltzmann's constant. The potential energy associated with the interaction of charged radicals or radicals possessing a dipole moment, with the surrounding medium must be assumed to be negligible for the above description to be valid. This assumption would be valid in those circumstances when charged groups do not reside on the surface of the (i, j)mer. The ( i , j ) mers can hence interact with the surrounding medium only when they come into physical contact with the molecules composing the media. The medium could also influence binding if it is possessing groups which interact with the Ab's and Ag's in which case binding of those groups with Ab's may be preferred over binding of Ab's and Ag's. Such effects can be included in the models by introducing specific non-linear reaction terms in the involved energy functions along with additional system equations and modifications in the involved probability functions p(ij "r) etc.

References [ 1 ] Chandrika B. and Joshi R.R. (1989), J. Theor. Biol. 141 (3) 285-302. [2] Chandrika B. and Joshi R.R. (1990), Int. J. Comp. Math. with Appl. 20 (8) 37-49. [3] Dar G. (1993), M.Sc. Home Paper Project Report (Advisor: R.R. Joshi), Department of Physics, lIT Bombay, [4] Hoebek J., Englebroghs Y., Chand S. and Strosberg A. (1987), Molecular Immunology, 24 (6) pp. 621-629. [5] Joshi R.R. (1990): PART - A FORTRAN subroutine for Constrained Integer Partitions (developed on CDC CYBER 180/840, IIT Bombay). [6] Joshi R.R. (1992), Comp. Math. Modelling 16 (10) 113-120. [7] Macken C. A. and Perelson A. S. (1985), Branching processses applied to cell surface aggregation phenomena, in: Lecture Notes in Biomathematics, Vol. 58 (Springer, Berlin). [8] Schittowski K. (1986), NLPQL - A FORTRAN subroutine solving nonlinear programming problems, Annals of O.R., 5. [9] Volkenstein M. V. (1983), Biophysics (MIR, Moscow). [10] Wang Z. X. (1990), J. Theor. Biol. 143, 445-463.