Anticooperativity in diffusion-controlled reactions with pairs of anisotropic domains: a model for the antigen–antibody encounter

Abstract

The encounter between anisotropic agents in diffusion-controlled reactions is a topic of very general relevance in chemistry and biology. Here we introduce a simplified model of encounter of an isotropic molecule with a pair of partially reacting agents and apply it to the encounter reaction between an antibody and its antigen. We reduce the problem to the solution of dual series relations, which can be solved iteratively, yielding the exact solution for the encounter rate constant at any desired order of accuracy. We quantify the encounter effectiveness by means of a simple indicator and show that the two binding centers systematically behave in an anticooperative fashion. However, we demonstrate that a reduction of the binding active sites allows the composite molecule to recover binding effectiveness, in spite of the overall reduction of the rate constant. In addition, we provide a simple formula that enables one to calculate the anticooperativity as a function of the size of the binding site for any values of the separation between the two active lobes and of the antigen size. Finally, some biological implications of our results are discussed.

Appendices

Appendix 1

In this appendix we illustrate the calculations leading to Eq. 31. Insertion of Eq. 30 into the general expression for the rate (Eq. 23) leads to the following equation:

$$ \begin{aligned} \kappa = & 4\pi Da{\sum\limits_{n = 0}^\infty {{\text{e}}^{{ - {\left( {2n + 1} \right)}\beta _{0} }} } } \\ & \quad \times {\left\{ {\sinh \beta _{0} \mathcal{I}_{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}} {\left( n \right)} + {\left( {2n + 1} \right)}\tanh {\left[ {{\left( {n + \frac{1} {2}} \right)}\beta _{0} } \right]}\mathcal{I}_{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} {\left( n \right)}} \right\}}, \\ \end{aligned} $$

(52)

where

$$ \mathcal{I}_{p} {\left( n \right)} = \frac{1} {{{\sqrt 2 }}}{\int\limits_0^\pi {\frac{{\sin \alpha P_{n} {\left( {\cos \alpha } \right)}}} {{{\left( {\cos \beta _{0} - \cos \alpha } \right)}^{p} }}{\text{d}}\alpha } }. $$

(53)

The integral $\mathcal{I}_{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} {\left( n \right)}$ can be calculated by recalling the identity given in Eq. 27. We obtain

$$ \begin{aligned} \mathcal{I}_{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} {\left( n \right)} = & {\sqrt 2 }{\sum\limits_{n = 0}^\infty {{\text{e}}^{{ - {\left( {m + \frac{1} {2}} \right)}\beta _{0} }} I_{{nm}} } } \\ = & \frac{{2{\sqrt 2 }}} {{2n + 1}}{\text{e}}^{{ - {\left( {n + \frac{1} {2}} \right)}\beta _{0} }} , \\ \end{aligned} $$

(54)

where we have made use of the orthonormality of Legendre polynomials,

$$ I_{{nm}} = {\int\limits_0^\pi {P_{n} {\left( {\cos \alpha } \right)}P_{m} {\left( {\cos \alpha } \right)}\sin \alpha \,{\text{d}}\alpha {\text{ = }}\frac{{\text{2}}} {{{\text{2n + 1}}}}\delta _{{n,m}} .} } $$

(55)

The integral $\mathcal{I}_{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}} {\left( n \right)}$ can be easily calculated by noting that

$$ \begin{aligned} \mathcal{I}_{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}} {\left( n \right)} = & - {\left( {\frac{2} {{\sinh \beta _{0} }}} \right)}\frac{{\partial \mathcal{I}_{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} {\left( n \right)}}} {{\partial \beta _{0} }} \\ = & \frac{{2{\sqrt 2 }}} {{\sinh \beta _{0} }}{\text{e}}^{{ - {\left( {n + \frac{1} {2}} \right)}\beta _{0} }} . \\ \end{aligned} $$

(56)

Insertion of Eqs. 54 and 56 into Eq. 52 leads to

$$ \kappa = 8\pi Da{\sum\limits_{n = 0}^\infty {{\text{e}}^{{ - {\left( {2n + 1} \right)}\beta _{0} }} } }{\left\{ {1 + \tanh {\left[ {{\left( {n + \frac{1} {2}} \right)}\beta _{0} } \right]}} \right\}}. $$

(57)

Finally, the expression for the rate constant (Eq. 31) is recovered by substituting into Eq. 57 the definitions of a and β₀ (Eqs. 13, 14).

Appendix 2

Insertion of Eq. 42 into Eq. 44 leads to the following equation:

$$ \begin{aligned} G^{*}_{0} = & {\sum\limits_{n = 0}^\infty {{\left( {2n + 1} \right)}{\left\{ {X_{n} - {\left( {q_{n} + X_{n} } \right)}\tanh {\left[ {{\left( {n + \frac{1} {2}} \right)}\beta _{0} } \right]}} \right\}}} }\mathcal{I}_{n} {\left( u \right)} \\ & \quad - \sin \beta _{0} {\sum\limits_{n = 0}^\infty {{\left( {q_{n} + X_{n} } \right)}\mathcal{J}_{n} {\left( u \right)}} }, \\ \end{aligned} $$

(58)

where

$$ \begin{aligned} {\mathcal{I}}_n \left( u \right) = & {\int\limits_u^\pi {\frac{{P_n \left( {\cos \alpha } \right)\sin \alpha }} {{\sqrt {\cos u - \cos \alpha } }}{\text{d}}\alpha .}} \\ { \mathcal{J}}_n \left( u \right) = & {\int\limits_u^\pi {\frac{{P_n \left( {\cos \alpha } \right)\sin \alpha }} {{\left[ {\cos \beta _0 - \cos \alpha } \right]\sqrt {\cos u - \cos \alpha } }}{\text{d}}\alpha .}} \\ \end{aligned} $$

(59)

These integrals can be solved by introducing the following identity (Sneddon 1966):

$$ \sqrt 2 \sum\limits_{n = 0}^\infty {\cos \left[ {\left( {n + \frac{1} {2}} \right)u} \right]P_n \left( {\cos \alpha } \right) = \frac{{\Theta \left( {\alpha - u} \right)}} {{\sqrt {\cos u - \cos \alpha } }}.} $$

(60)

The integral $\mathcal{I}_n \left( u \right)$ can be easily evaluated using Eq. 60 and recalling the orthonormality condition of Legendre polynomials. We get

$$ \begin{aligned} {\mathcal{I}}_n \left( u \right) = & \sqrt 2 \sum\limits_{n = 0}^\infty {\cos \left[ {\left( {m + \frac{1} {2}} \right)u} \right]I_{nm} } \\ = & \frac{{2\sqrt 2 }} {{2n + 1}}\cos \left[ {\left( {n + \frac{1} {2}} \right)u} \right]\;. \\ \end{aligned} $$

(61)

Insertion of the identity (Eq. 60) into the definition of $\mathcal{J}_n \left( u \right)$ gives

$$ \mathcal{J}_{n} {\left( u \right)} = {\sqrt 2 }{\sum\limits_{n = 0}^\infty {\cos {\left[ {{\left( {m + \frac{1} {2}} \right)}u} \right]}\Gamma _{{nm}} } }, $$

(62)

with

$$ \Gamma _{nm} = \frac{1} {2}\int\limits_u^\pi {\frac{{P_n \left( {\cos \alpha } \right)P_m \left( {\cos \alpha } \right)\sin \alpha }} {{\cos \beta _0 - \cos \alpha }}{\text{d}}\alpha .} $$

(63)

The integral in Eq. 63 can be solved by using Eq. 7.224.5 from Gradshteyn and Ryzhik (1980). The result is Eq. 47. Then, insertion of Eq. 58 into Eq. 43 leads to the calculation of two integrals of the type

$$ {\int\limits_u^\pi {\cos {\left[ {{\left( {p + \frac{1} {2}} \right)}u} \right]}\cos {\left[ {{\left( {q + \frac{1} {2}} \right)}u} \right]}{\text{d}}u = \frac{\pi } {2}S_{{pq}} } }, $$

(64)

with S _pq defined as in Eq. 46.

Taking into account expressions Eqs. 58 and 64, it is a simple matter of algebraic rearrangement to put Eq. 43 in the form of the infinite set of Eq. 45.

As a check of consistency, we observe that the case of entirely absorbing Fabs may be easily recovered. In this case one has S _mn =M _mn =0 for all m, n, and hence X _n =0 for all n. Consequently, Eq. 40 reduces to Eq. 29, and the solution (Eq. 30) is recovered along with the rate constant (Eq. 31) (see also Eq. 48). On the other hand, the solution to the Laplace problem must vanish in the limit α₀→0, where the two Fabs become entirely reflecting. In this case one has S _mn →δ _mn , and hence the problem reduces to solving the following infinite homogeneous system in the variables X _n +q _n (see Eq. 40),

$$ {\sum\limits_{m = 0}^\infty {\Xi _{{nm}} {\left( {X_{m} + q_{m} } \right)}} } = {\sum\limits_{m = 0}^\infty {{\left( {\delta _{{nm}} + M_{{nm}} } \right)}{\left( {X_{m} + q_{m} } \right)} = 0} }, $$

(65)

whose only solution is the trivial one X _n +q _n =0 ∀ n, being det(Ξ)≠0 at each order of truncation (which can easily be proved numerically to be the case). This proves that the solution to the Laplace problem is indeed the trivial one, and that the rate constant vanishes (see also Eq. 48).

Appendix 3

In this appendix we discuss in some detail the convergence properties associated with truncation of the infinite set of equations (Eq. 45).

If we indicate with N _t (θ₀,χ) the number of equations retained in the infinite system (Eq. 45) for a given choice of the parameters (θ₀,χ) within the intervals of interest, we find that a relative error δX _n /X _n <0.1% ∀n may be obtained by keeping in the worst case up to order N _t =100 equations. In the calculation of the norm of M, the sum in the definition (Eq. 46) has been truncated at a fixed number of terms N*, independent of N. To understand whether that is legitimate, let us consider that term in more detail. Since |S _mn |<1∀m,n, we have

$$ \begin{aligned} {\left| {{\sqrt {\chi ^{2} - 1} }{\sum\limits_{m = 0}^{N_{t} } {{\sum\limits_{k = 0}^{N*} {\Gamma _{{mk}} S_{{kn}} } }} }} \right|} < & {\sqrt {\chi ^{2} - 1} }{\sum\limits_{m = 0}^{N_{t} } {{\left| {{\sum\limits_{k = 0}^{N*} {\Gamma _{{mk}} } }} \right|}} } \\ < & {\sqrt {\chi ^{2} - 1} }{\sum\limits_{m = 0}^{N_{t} } {{\left| {{\sum\limits_{k = 0}^{N*} {\Gamma _{{mk}} } }} \right|}\underline{\underline {{\text{def}}}} \gamma {\left( {N*} \right)}.} } \\ \end{aligned} $$

(66)

The quantity γ(N*) is plotted in Fig. 9 for three values of χ, with two choices of the truncation order N _t each. It is very clear that such an indicator converges rapidly for all values of χ. The plot confirms that the correct value of N* in order to assure convergence increases with the order of truncation N _t . In particular, we have N*≈N _t . Hence, the greater precision we require in the numerical solution for the coefficients X _n , the greater must be N _t , and consequently the greater N*. Operatively, we can thus safely require the specified accuracy to give the rules for both truncations.

Fig. 9

γ(N*). Solid lines, N _t =80 Dashed lines, N _t =100