# Interplay of receptor memory and ligand rebinding

Thorsten Prüstel  Laboratory of Systems Biology
National Institute of Allergy and Infectious Diseases
National Institutes of Health  Martin Meier-Schellersheim  Laboratory of
Systems Biology
National Institute of Allergy and Infectious Diseases
National Institutes of Health

###### Abstract

Rapid rebinding of molecular interaction partners that are in close proximity
after dissociation leads to a dissociation and association kinetics that can
profoundly differ from predictions based on bulk reaction models. The cause of
this effect can be traced back to the non-Markovian character of the ligand’s
rebinding time probability density function, reflecting the fact that, for a
certain time span, the ligand still ’remembers’ the receptor it was bound to
previously. In this manuscript, we explore the consequences of the hypothesis
that initial binding and consecutive rebinding give rise to a bond lifetime
density that is non-Markovian as well. We study the combined effect of the two
non-Markovian waiting time probability densities and show that even for very
short times the decay of the fraction of occupied receptors deviates from an
exponential. For long times, dissociation is slower than an exponential and
the fate of the the steady-state bound receptor fraction critically depends on
the extent of the deviation, relative to the rebinding time density, from the
Markovian limit: The population of occupied receptors may either decay
completely or assume a non-vanishing value for small and strong deviations,
respectively. Furthermore, we point out the important role played by
fractional calculus and demonstrate that the short- and long-time dynamics of
the occupied receptors can be naturally expressed as well as easily obtained
in terms of fractional differential equations involving the Riemann-Liouville
derivative. Our analysis shows that cells may exploit receptor memory as
mechanism to dynamically widen the range of potential response-patterns to a
given signal.

11footnotetext: Email<EMAIL_ADDRESS><EMAIL_ADDRESS>
##  1 Introduction

Growing evidence [8, 9, 25, 18] suggests that a ligand’s diffusive motion
results in a receptor-ligand interaction pattern and a network response
further downstream that can deviate substantially from the picture based on
deterministic first-order kinetics. A major cause of these differences has
been identified as the non-Markovian rebinding time probability density
function (pdf) that arises in stochastic, spatially-resolved accounts of the
ligand’s motion [25, 18]. This rebinding time pdf may be considered as a
’deformed’ version of the memoryless exponential waiting time pdf that
corresponds to the first order kinetics of a deterministic mean-field
description. In fact, the rebinding time pdf $\psi_{\text{reb}}(t)$ for
1-dimensional diffusive motion in the presence of a partially absorbing
boundary takes in the Laplace domain the form (see appendix A)

$\mathcal{L}[\psi_{\text{reb}}(t)](s):=\tilde{\psi}_{\text{reb}}(s):=\int^{\infty}_{0}\psi_{\text{reb}}(t)e^{-st}dt=\frac{\kappa_{D}^{1/2}}{\kappa_{D}^{1/2}+s^{1/2}},$
(1)

where $\kappa_{D}=\kappa^{2}_{a}/D$ is a parameter whose strength is
determined by the ratio of the intrinsic association constant $\kappa_{a}$ to
the diffusion constant $D$. Note that $\kappa_{D}$ has the dimension of an
inverse time.

By contrast, the exponential waiting time pdf that is commonly employed to
describe the dissociation reaction (and association as well in a non-spatial
deterministic approach),

$\psi(t)=\kappa_{d}e^{-\kappa_{d}t},$ (2)

reads in the Laplace domain as

$\tilde{\psi}(s)=\frac{\kappa_{d}}{\kappa_{d}+s},$ (3)

where $\kappa_{d}$ refers to the intrinsic dissociation constant. Both density
functions (Eqs. (1), (3)) are representatives of a family of pdf that are
parametrized by a parameter $0<\sigma\leq 1$ [2, 10, 21]

$\tilde{\psi}(s)=\frac{\kappa^{\sigma}}{\kappa^{\sigma}+s^{\sigma}}.$ (4)

Any PDF with $\sigma<1$ describes memory effects: Because a ligand, that just
dissociated from a receptor, will remain close to that receptor for a certain
time span it is more likely to bind the same receptor again. The spatial
correlations between receptor and dissociated ligand translate to a time
evolution of the ligand’s (survival) probability that is non-local in time and
hence give rise to a dynamics that can be naturally cast as a fractional
differential equation. Fractional calculus [19, 17] plays a prominent role in
anomalous diffusion and non-equilibrium statistical mechanics of complex
systems [14, 23, 16, 24, 12, 26], but it also naturally arises in the context
of normal diffusion. To see this in more detail, let us recall that the
rebinding PDF $\psi_{\text{reb}}(t)$ of a single ligand is related to the
survival probability of a molecule that undergoes a 1-dimensional diffusive
motion in the presence of a partially absorbing boundary condition (BC) at
$z_{0}=0$ by (cf. Appendix A)

$\psi_{\text{reb}}(t)=-\frac{\partial S_{\text{rad}}(t|z_{0}=0)}{\partial t},$
(5)

Henceforth, we will use the notation $S(t):=S_{\text{rad}}(t|z_{0}=0)$ for
convenience. From Eqs. (1), (5) and (45) we can conclude that

$1-s\tilde{S}(s)=\kappa^{1/2}_{D}s^{1/2}\tilde{S}(s).$ (6)

This relation reads in the time domain as

$\frac{\partial S(t)}{\partial t}=-\kappa^{1/2}_{D}D^{1-1/2}_{t}S(t),$ (7)

where $D^{1-\sigma}_{t}$ denotes the Riemann-Liouville (R-L) fractional
differential operator that is defined as [19, 17, 14]

$_{0}D^{1-\sigma}_{t}f(t):=D^{1-\sigma}_{t}f(t):=\frac{1}{\Gamma(\sigma)}\frac{\partial}{\partial
t}\int^{t}_{0}\frac{f(\tau)}{(t-\tau)^{1-\sigma}}d\tau,\quad 0<\sigma<1.$ (8)

The definition of the R-L derivative (Eq. (8)) makes it evident that the time
evolution of $S(t)$ (Eq. (7)) is indeed governed by an equation that is non-
local in time. Eq. (7) generalizes the well-known Markovian time evolution

$\frac{\partial S(t)}{\partial t}=-\kappa_{D}S(t)$ (9)

that leads to a Poissonian form of the rebinding pdf. Solutions of evolution
equations that take the form of a fractional relaxation law [7, 14, 15]

$\frac{\partial\Phi(t)}{\partial t}=-\kappa^{\sigma}D^{1-\sigma}_{t}\Phi(t)$
(10)

are given in terms of Mittag-Leffler (M-L) functions [5, 13]
$\Phi(t)=E_{\sigma}[-(\kappa t)^{\sigma}]$. M-L functions may be thought of a
generalized exponentials that are defined by the series expansion

$E_{\alpha}(-x)=\sum^{\infty}_{n=0}\frac{(-x)^{n}}{\Gamma(n\alpha+1)},$ (11)

which makes it evident that for $\alpha=1$ one recovers an exponential
$E_{1}(-t)=\exp(-t)$. The Laplace transform is

$\mathcal{L}\bigg{[}E_{\sigma}(-(\kappa
t)^{\sigma})\bigg{]}=\frac{1}{s}\frac{s^{\sigma}}{s^{\sigma}+\kappa^{\sigma}}$
(12)

and hence it follows that the pdf Eq. (4) is the Laplace transform of the
negative time derivative of the Mittag-Leffler (M-L) function
$E_{\sigma}[-(\kappa t)^{\sigma}]$ [10]:

$\psi(t)=-\frac{\partial E_{\sigma}[-(\kappa t)^{\sigma}]}{\partial t}.$ (13)

Because $E_{1/2}[-t^{1/2}]=\exp(t)\text{erfc}(\sqrt{t})$ we immediately obtain
from Eq. (7) the known expression for $S(t)$, cf. Appendix A, Eq. (46). For
small arguments, the M-L function behaves like a Kohlrausch-Williams-Watts
(KWW) function, i.e. a stretched exponential
$E_{\alpha}[-t^{\alpha}]\sim\exp(-t^{\alpha})$ [14, 15] (cf. Eq. (11)), while
for large arguments it decays like an inverse power law. More precisely, one
has [13, 14]

$\displaystyle E_{\alpha}[-t^{\alpha}]$ $\displaystyle=$
$\displaystyle\frac{t^{-\alpha}}{\Gamma(1-\alpha)}-\frac{t^{-2\alpha}}{\Gamma(1-2\alpha)}+\ldots\quad
t\rightarrow\infty.$ (14)

We have seen that fractional evolution equations (Eq. (7)) that reflect
deviations from a Markovian waiting time pdf naturally appear in the
decription of ligand rebinding, even in the case of normal diffusion. However,
the reverse process, dissociation is usually taken into account by a
memoryless pdf for the bond lifetime, even in particle-based stochastic
approaches (however, note, for instance, Refs. [7, 11]). Here, we suggest to
abandon the assumption of the Markovian property. A deviation from a
memoryless bond lifetime pdf is motivated by several lines of experimental
evidence suggesting that the receptor state does not remain unaltered upon
initial rebinding, for instance due to receptor deformation or conformational
changes, the formation of signaling intermediates, involvement of the actin
skeleton and cluster formation and other mechanisms [7, 11, 27]. In the
picture that we will consider, the bound state undergoes a diffusion-like
’motion’ in an abstract internal ’space’ whose properties are dynamically
defined by the local biochemistry near the receptor. We will show that the
interplay of these two different diffusive processes alters substantially the
apparent dissociation kinetics and can even lead to behavior that has no
counterpart in models that only take into account either the non-Markovian
character of the ligand rebinding or the bond lifetime alone.

Let us recall how an altered receptor state may lead to a bond lifetime pdf
that is sufficiently well described by Eq. (4) [2, 7]. To this end, one
assumes that the receptor modifications imply a bond lifetime pdf that is
given by a superposition of exponentials

$\psi(t)=\sum_{n}\phi_{n}e^{-\kappa_{n}t}.$ (15)

One further assume that the sum can be approximated by an integral

$\psi(t)=\int^{\infty}_{0}\phi(\kappa)e^{-\kappa t}d\kappa.$ (16)

Then, using mild assumptions, one can derive (cf. Appendix B) that,
asymptotically, the pdf behaves like a power-law

$\psi(t)\propto\frac{\Gamma(1+\sigma)}{t^{1+\sigma}}.$ (17)

Employing Tauberian theorems [6, 12] one can show that

$\tilde{\psi}(s)\sim 1-(\tau s)^{\sigma},$ (18)

where $\tau$ denotes a constant having dimension of time. Now, for large
times, i.e. small $s$ or for small $\tau$ we finally may rewrite Eq. (18) as
Eq. (4), identifying $\tau^{-1}=\kappa$.

Therefore, we will henceforth assume that the bond lifetime pdf is indeed
given by Eq. (4). We consider the following system: Ligands diffuse in a 3
dimensional space that is semi-infinite, bounded by a two-dimensional plane
homogeneously covered with receptors. We adopt the self-consistent stochastic
mean-field theory approach of Ref. [8] that allows to track the motion of the
ligand accurately, while the receptors are described in a non-spatial fashion
by the fraction of occupied receptors. We point out that the formalism is
flexible enough to include also a spatially-resolved description of the
receptors [9]. Here, however, we aim at emphasizing the consequences of a non-
Markovian bond lifetime density and its interplay with the rebinding pdf and
therefore limit our analyses to a non-spatial receptor description.

##  2 Theory

First, we focus on the Markovian case [8]. We consider a membrane that extends
infinitely in the $x-y$ plane ($z=0$) and that is uniformly covered with
receptors. We assume that all ligands are initially bound. The fraction of
occupied receptors is denoted by $p(t)=\mathcal{R}(t)/\mathcal{R}_{0}$ where
$\mathcal{R}(t)$ and $\mathcal{R}_{0}$ refer to the density of bound receptors
at time $t$ and the constant mean receptor surface density per unit area,
respectively. Its time evolution is governed by [8]

$\frac{dp(t)}{dt}=-k_{\text{off}}p(t)+k_{\text{on}}\rho(0,t)[1-p(t)],$ (19)

where $\rho(z,t)$ describes the concentration of ligands at a distant $z$ from
the membrane. To solve Eq. (19) one has to express the ligand density
$\rho(0,t)$ and hence the rebinding rate

$\gamma(t)=k_{\text{on}}\rho(0,t)[1-p(t)]$ (20)

in terms of $p(t)$. To this end, we follow the self-consistent mean-field
theory approach described in Refs. [8] and assume that $p(0)\ll 1$ and hence
$p(t)\ll 1$ for all times $t$ so that the rebinding rate reduces to
$\gamma(t)=k_{\text{on}}\rho(0,t)$. We now aim at expressing $\gamma(t)$ in an
alternative form that incorporates spatially-resolved information about the
diffusive motion of the ligand. The fraction of occupied receptors which is
converted to the unbound state between time $\tau$ and $\tau+d\tau$ is given
by $k_{\text{off}}p(\tau)d\tau$. A dissociated ligand begins to undergo an
effectively one-dimensional diffusive motion that starts at the mebrane $z=0$
between time $\tau$ and $\tau+d\tau$. Let $\mathcal{S}(t-\tau|z=0)$ denote the
probability that a ligand is still unbound when the time span $t-\tau$ has
elapsed since the dissociation. Then, the rate of absorption is given by by
the rebinding time pdf (see Appendix C)

$\psi_{\text{reb}}(t):=R_{\text{rad}}(t-\tau|z_{0}=0):=-\frac{\mathcal{S}(t-\tau|z_{0}=0)}{\partial
t}$ (21)

and it follows that the rebinding rate can be written as convolution

$\gamma(t)=k_{\text{off}}\int^{t}_{0}d\tau p(\tau)R_{\text{rad}}(t-\tau|0).$
(22)

Note that, despite the formal resemblance of Eqs. (5) and (21), we emphasize
that the context of both relations is quite different, because here we
consider a many-molecule system, instead of a single ligand-receptor pair.
Also, the self-consistent mean-field theory approach is more flexible and can
be applied to more general cases [9].

As it stands, Eq. (19) describes memoryless dissociation based on the
exponential waiting time pdf Eq. (2). To introduce a non-Markovian intrinsic
bond lifetime, we perform the substitution

$k_{\text{off}}\rightarrow s\frac{\tilde{\psi}(s)}{1-\tilde{\psi}(s)}$ (23)

in the Laplace domain, where $\tilde{\psi}(s)$ is given by Eq. (4) with
$\kappa=k_{\text{off}}$. Clearly, for $\sigma=1$, one recovers
$s\tilde{\psi}(s)/[1-\tilde{\psi}(s)]=k_{\text{off}}$. The substitution rule
Eq. (23) has been utilized before [2, 21]. However, Refs. [2, 21] have not
identified the action of the operation of Eq. (23) in the time domain. Using
[14]

$\mathcal{L}[D^{1-\sigma}_{t}f(t)](s)=s^{1-\sigma}\tilde{f}(s),$ (24)

it turns out that it corresponds to

$-k_{\text{off}}p(t)\longrightarrow-k^{\sigma}_{\text{off}}D^{1-\sigma}_{t}p(t)$
(25)

where $D^{1-\sigma}_{t}p(t)$ refers to the R-L derivative (Eq. (8)). Now,
combining Eqs. (12), (19), (22), (24), (25) and (59) we obtain

$\tilde{p}(s)=s^{-1}p(0)-k^{\sigma}_{\text{off}}s^{-\sigma}\frac{s^{1/2}}{s^{1/2}+\kappa^{1/2}_{D}}\tilde{p}(s),$
(26)

or, equivalently,

$p(t)=p(0)-k^{\sigma}_{\text{off}}D^{1-\sigma}_{t}\bigg{\\{}E_{1/2}[-(\kappa_{D}t)^{1/2}]\ast
p(t)\bigg{\\}},$ (27)

where $\ast$ denotes a convolution integral.

Eq. (27) represents a natural generalization of a rate equation approach,
abandoning the assumptions of both a memoryless rebinding and dissociation
waiting time pdf. In fact, considering the limit $\sigma\rightarrow 1$ and
substituting $E_{1/2}[-(\kappa_{D}t)^{1/2}]\rightarrow
E_{1}[-(\kappa_{D}t)^{1}]=\exp(-\kappa_{D}t)$, we obtain from Eq. (27) the
deterministic Markovian mean-field result

$p(t)=p(0)\bigg{[}e^{-(\kappa_{D}+k_{\text{off}})t}\frac{k_{\text{off}}}{\kappa_{D}+k_{\text{off}}}+\frac{\kappa_{D}}{\kappa_{D}+k_{\text{off}}}\bigg{]}.$
(28)

Next, we turn to the general case and focus on the short and long time limit.
We will show that in most cases the dynamics is described by the same
fractional relaxation law (Eq. (10)), only the order of the R-L derivative
changes from case to case. Hence, as a benefit, the solutions can be virtualy
read off, a fact which underlines the power and elegance of the chosen
formalism.

##  3 Results

###  3.1 Solution for short times $t\rightarrow 0$

For very short times, the effect of the rebinding should be negligible. This
expectation is well reflected by Eq. (27): Because short times imply that
$E_{1/2}[-(\kappa_{D}t)^{1/2}]\rightarrow 1$, it follows that the equation of
motion assumes the form of a fractional relaxation law,

$\frac{dp(t)}{dt}=-k^{\sigma}_{\text{off}}D^{1-\sigma}_{t}p(t),$ (29)

for which we can swiftly read off the solution (cf. Eqs. (10) and (11))

$\displaystyle p(t)$ $\displaystyle=$ $\displaystyle
p(0)E_{\sigma}[-(k_{\text{off}}t)^{\sigma}]$ (30) $\displaystyle=$
$\displaystyle
p(0)\bigg{[}1-\frac{(k_{\text{off}}t)^{\sigma}}{\Gamma(1+\sigma)}+\frac{(k_{\text{off}}t)^{2\sigma}}{\Gamma(1+2\sigma)}-\ldots\bigg{]}\quad\text{for}\,t\rightarrow
0.$ (31)

Note that to arrive at Eq. (29), we invoked the commutation relation
$[D^{1-\sigma}_{t},\partial/\partial t]f(t)=0$ that does hold if $f(0)=0$.
Since in the case considered here we have $f(t)=\int^{t}_{0}p(\tau)d\tau$, the
condition is fullfilled. As we know from the behavior of the M-L function (and
can see from Eq. (31)), for short times the fraction of occupied receptors
decays like a stretched exponential $p(t)\sim
p(0)\exp[-(k_{\text{off}}t)^{\sigma}]$.

We emphasize that the solution given by Eq. (30) is not only valid for short
times, but in addition in parameter regimes in which
$\kappa^{1/2}_{D}=\mathcal{R}_{0}k_{\text{on}}/\sqrt{D}$ becomes small, i.e.
when either the receptor density is small or when rebinding effects are weak,
i.e. the ratio of on-rate to diffusion constant becomes small. Because for
$\sigma=1$ the corresponding M-L function in Eq. (30) is a decaying
exponential $E_{1}[-(k_{\text{off}}t)^{1}]=\exp(-k_{\text{off}}t)$, we recover
the result from [8] for short times. However, here we see that abandoning the
assumption of a memoryless bond lifetime leads to an decay that deviates from
an exponential even for short times or for small $\kappa_{D}$.

###  3.2 Solutions for long times $t\rightarrow\infty$

Now we will show that the combined effect of rebinding and receptor memory
gives rise to behavior that has no counterpart in cases in which one uses only
one of the two non-Markovian densities. The long time limit, or equivalently
$s\rightarrow 0$, implies that (cf. Eq. (12))

$\mathcal{L}\bigg{\\{}E_{1/2}[-(\kappa_{D}t)^{1/2}]\bigg{\\}}\rightarrow\frac{1}{\kappa^{1/2}_{D}s^{1/2}}.$
(32)

As a result, Eq. (27) again takes the form of a fractional relaxation law, as
in the case of short times, but the order of the R-L derivative has changed
(cf. with Eq. (29))

$\frac{dp(t)}{dt}=-\frac{k^{\sigma}_{\text{off}}}{\kappa^{1/2}_{D}}D^{1-\sigma+1/2}_{t}p(t).$
(33)

Here, we again emphasize that the found relaxation law is not only valid for
long times, but also for regimes in which $\kappa_{D}$ becomes large, i.e.
where rebinding effects becomes strong. Furthermore, the form of Eq. (33)
makes it evident that there is now a crucial difference whether $\sigma<1/2$
or $\sigma>1/2$ i.e. whether the bond lifetime is to a larger extent non-
Markovian than the rebinding time pdf or not. In this sense, diffusion sets
the scale for what should be considered a strong or weak deviation from
memoryless behavior. Note that this scale is not absolute: If the ligand
undergoes anomalous diffusion [14, 16, 23, 24], the scale will not be $1/2$
anymore.

We now consider several special cases separately. First, let us focus on the
Markovian limit $\sigma=1$. Then, we immediately obtain from Eq. (33)

$p(t)=E_{1/2}\bigg{[}-\bigg{(}\frac{k^{2}_{\text{off}}}{\kappa_{D}}t\bigg{)}^{1/2}\bigg{]}=\exp\bigg{[}\bigg{(}\frac{k^{2}_{\text{off}}}{\kappa_{D}}t\bigg{)}\bigg{]}\text{erfc}\bigg{[}\sqrt{\frac{k^{2}_{\text{off}}}{\kappa_{D}}t}\bigg{]},$
(34)

which is the result given in Ref. [8, 9], as expected. In particular, we
observe a power law decay for long times $p(t\rightarrow\infty)\sim
1/\sqrt{\pi t}$.

Next, we analyze the case $1>\sigma>1/2$. It follows that $\sigma-1/2>0$ and
hence we can read off again immediately the solution from Eq. (33)

$\displaystyle p(t)$ $\displaystyle=$ $\displaystyle
p(0)E_{\sigma-1/2}\bigg{[}-k^{\sigma}_{\text{off}}/\kappa^{1/2}_{D}t^{\sigma-1/2}\bigg{]}=$
(35) $\displaystyle
p(0)\bigg{[}\frac{\kappa^{1/2}_{D}}{k^{\sigma}_{\text{off}}\Gamma[1-(\sigma-1/2)]t^{\sigma-1/2}}-\frac{\kappa_{D}}{k^{2\sigma}_{\text{off}}\Gamma[1-2(\sigma-1/2)]t^{2\sigma-1}}+\ldots\bigg{]},\qquad\quad$

where we employed the expansion of the M-L function for large arguments Eq.
(14). We note that the fraction of occupied receptors will completely decay
for long times, as in the Markovian case $\sigma=1$. However, it can also be
seen from Eq. (35) that for general $1/2<\sigma<1$ the decay can become
arbitrarily slow for $\sigma\rightarrow 1/2$ and hence $\sigma=1/2$ seems to
play a special role that we are going to consider now.

As previously, the asymptotic solution $p(t\rightarrow\infty)$ for the
limiting case $\sigma=1/2$ can easily be obtained from the general relaxation
law Eq. (33) and using $E_{0}(-x)=1/(1+x)$

$p(t\rightarrow\infty)=p(0)\frac{\kappa^{1/2}_{D}}{\kappa^{1/2}_{D}+k^{1/2}_{\text{off}}}.$
(36)

Thus, we find that the steady-state receptor occupancy is not vanishing in the
long time limit, in stark contrast to the case of an exponential decay of the
intrinsic bond lifetime. We point out that Agmon and Weiss found a
corresponding behavior for the case of a single pair in one dimension [2]. But
as mentioned before, the formalism adopted here is more general [8, 9].
Moreover, we establish a connection to 3D/2D membrane binding as well as
fractional calculus and the crucial role of the fractional relaxation law.

Finally, we turn to the case $\sigma<1/2$. This is the only case for which we
cannot read off the solution immediately. The reason is that the nature of the
R-L derivative changes qualitatively. In fact, let be $\alpha=1/2-\sigma$,
then $\alpha>0$ and one has [14]

$D^{1+\alpha}_{t}p(t)=\frac{1}{\Gamma(1-\alpha)}\frac{\partial^{2}}{\partial
t^{2}}\int^{t}_{0}\frac{p(\tau)}{(t-\tau)^{\alpha}}d\tau.$ (37)

Therefore, we Laplace transform Eq. (33) (or, alternatively, start from (26))
and obtain

$\displaystyle\tilde{p}(s)$ $\displaystyle=$
$\displaystyle\kappa^{1/2}_{D}p(0)\frac{s^{\sigma-1}}{\kappa^{1/2}_{D}s^{\sigma}+k^{\sigma}_{\text{off}}s^{1/2}}$
(38) $\displaystyle=$
$\displaystyle\frac{p(0)}{s}\bigg{[}1-\frac{1}{1+\kappa^{1/2}_{D}/k^{\sigma}_{\text{off}}s^{\sigma-1/2}}\bigg{]}.$

The corresponding expression in the time domain follows from Eq. (12)

$p(t)=p(0)\bigg{\\{}1-E_{1/2-\sigma}\bigg{[}-\kappa^{1/2}_{D}/k^{\sigma}_{\text{off}}t^{1/2-\sigma}\bigg{]}\bigg{\\}}$
(39)

In particular, we find that

$\lim_{t\rightarrow\infty}p(t)=p(0)$ (40)

the initial fraction of occupied receptors is restored completely.

## Appendix A

We consider a molecule undergoing 1-dimensional diffusive motion in the
presence of a partially absorbing BC. The pdf $g_{\text{rad}}(z,t|z_{0})$ that
yields the probability $g_{\text{rad}}(z,t|z_{0})dz$ to find the molecule
between $z$ and $z+dz$ at time $t$, provided that it was initially located at
$z_{0}$ at time $t_{0}=0$, is the GF of the diffusion equation

$\frac{\partial g_{\text{rad}}(z,t|z_{0})}{\partial
t}=D\frac{\partial^{2}g_{\text{rad}}(z,t|z_{0})}{\partial z^{2}},$ (41)

subject to the radiation BC [4, 22]

$D\frac{\partial g_{\text{rad}}(z,t|z_{0})}{\partial
z}|_{z=0}=\kappa_{a}g_{\text{rad}}(z=0,t|z_{0}).$ (42)

The GF takes in the Laplace domain the form [3, p. 358]

$\tilde{g}_{\text{rad}}(z,s|z_{0})=\frac{e^{-\sqrt{s/D}|z-z_{0}|}}{2\sqrt{Ds}}+\frac{e^{-\sqrt{s/D}(z+z_{0})}}{2\sqrt{Ds}}-\frac{\kappa_{a}e^{-\sqrt{s/D}(z-z_{0})}}{D\sqrt{s}[\sqrt{s}+\kappa_{a}/\sqrt{D}]}.$
(43)

The survival probability $S_{\text{rad}}(t|z_{0})$ is defined as [22]

$S_{\text{rad}}(t|z_{0})=\int^{\infty}_{0}g_{\text{rad}}(z,t|z_{0})dz,$ (44)

and hence one finds

$\tilde{S}_{\text{rad}}(s|z_{0}=0)=\frac{1}{s}\frac{s^{1/2}}{s^{1/2}+\kappa^{1/2}_{D}},$
(45)

where $\kappa_{D}=\kappa^{2}_{a}/D$. Eq. (45) becomes in the time domain [1,
Eq. (29.3.43)]

$S_{\text{rad}}(t|z_{0}=0)=\exp(\kappa_{D}t)\text{erfc}(\sqrt{\kappa_{D}t}).$
(46)

Furthermore, from Eqs. (41) and (42) one may derive the relation

$-\frac{S_{\text{rad}}(t|z_{0})}{\partial
t}=\kappa_{a}g_{\text{rad}}(z=0,t|z_{0}).$ (47)

Finally, using Eqs. (43) and (47) we obtain

$\tilde{\psi}_{\text{reb}}(s)=\mathcal{L}\bigg{[}-\frac{\partial
S_{\text{rad}}(t|z_{0}=0)}{\partial
t}\bigg{]}=\frac{\kappa^{1/2}_{D}}{\kappa^{1/2}_{D}+s^{1/2}}$ (48)

## Appendix B

In this Appendix we recall [2, 7, 12] how the form of the density
$\phi(\kappa)$, that appears in Eq. (16), can be derived from rather mild
assumptions. First, one assumes that a typical rate $\kappa$ is characterized
by an Arrhenius factor

$\kappa\propto e^{-\frac{E}{kT}},$ (49)

where $k,T$ refer to the Boltzmann constant and the absolute temperature,
respectively. The activation energies $E$ themselves are assumed to be
distributed according to an Arrhenius factor too

$\phi(E)\propto e^{-\frac{E}{E_{0}}},$ (50)

where $E_{0}$ denotes a energy reference scale. Using Eq. (49), one may
express $E$ by $\kappa$ and substitute the obtained expression in Eq. (50) to
find

$\phi(\kappa)\propto\kappa^{\sigma},$ (51)

where $\sigma:=kT/E_{0}$. Eq. (16) defines a Laplace transform where $\kappa$
plays the role of the time variable. Hence, knowing Eq. (51), one may employ
$\mathcal{L}[t^{\alpha-1}]=\Gamma(\alpha)/s^{\alpha},\,\alpha>0$ [1, Eq.
(29.3.7)] to finally arrive at Eq. (17).

## Appendix C

We consider first a single ligand-receptor pair. The rate of recombination
$R_{\text{rad}}(t|z_{0})=-\partial S_{\text{rad}}(t|z_{0})/\partial t$ can be
written as convolution [20]

$R_{\text{rad}}(t|z_{0})=\Theta
R_{\text{abs}}(t|z_{0})+(1-\Theta)\int^{t}_{0}R_{\text{abs}}(\tau|z_{0})R_{\text{rad}}(t-\tau|\Delta)d\tau\quad\text{for}\,\Delta\rightarrow
0$ (52)

where $R_{\text{abs}}(t|z_{0})$ refers to the rate of first encounters (first-
passage time pdf)

$R_{\text{abs}}(t|z_{0})=-\frac{S_{\text{abs}}(t|z_{0})}{\partial t},$ (53)

and $S_{\text{abs}}(t|z_{0})$ denotes the survival probability in the presence
of an absorbing BC. Furthermore, $\Delta$ is a microscopic length scale and
$\Theta$ denotes the probability that a single molecule pair reaction takes
place when they encounter. However, here we are interested in a situation
where we consider many receptors. Eq. (52) remains valid [8, 9], but the
probability $\Theta$ has to be adapted properly. Following Refs. [8, 9], we
define

$\Theta=\frac{\mathcal{R}_{0}k_{\text{on}}\Delta}{D}.$ (54)

Next, we Laplace transform Eq. (52) to arrive at

$\tilde{R}_{\text{rad}}(s|z_{0})=\Theta\tilde{R}_{\text{abs}}(s|z_{0})+(1-\Theta)\tilde{R}_{\text{abs}}(s|z_{0})\tilde{R}_{\text{rad}}(s|\Delta).$
(55)

Setting $z_{0}=\Delta$, we obtain from Eq. (55) an expression for
$\tilde{R}_{\text{rad}}(s|z_{0}=\Delta)$ and substite it back in Eq. (55) to
find

$\tilde{R}_{\text{rad}}(s|z_{0})=\frac{\Theta\tilde{R}_{\text{abs}}(s|z_{0})}{1-(1-\Theta)\tilde{R}_{\text{abs}}(s|\Delta)}.$
(56)

Using $\tilde{R}_{\text{abs}}(s|z_{0})=e^{-\frac{z_{0}}{\sqrt{D}}\sqrt{s}}$
[8] and Taylor-expanding up to first order in $\Delta$ yields

$\tilde{R}_{\text{rad}}(s|z_{0})=\frac{\Theta\tilde{R}_{\text{abs}}(s|z_{0})}{\Theta+(1-\Theta)\Delta/\sqrt{D}s^{1/2}}$
(57)

Finally, we take the limit $\Delta\rightarrow 0$ (cf. Eq. (54))

$\lim_{\Delta\rightarrow
0}\frac{\Delta}{\Theta}=\frac{D}{k_{\text{on}}\mathcal{R}_{0}}$ (58)

to arrive at

$\tilde{\psi}_{\text{reb}}(s)=\tilde{R}_{\text{rad}}(s|z_{0}=0)=\frac{1}{1+\kappa^{-1/2}_{D}s^{1/2}},\quad\kappa^{1/2}_{D}:=\mathcal{R}_{0}k_{\text{on}}/\sqrt{D}.$
(59)

In particular, this result means that the rate of recombination is the
negative time derivative of the M-L function
$E_{1/2}[-\mathcal{R}_{0}k_{\text{on}}/\sqrt{D}t^{1/2}]$.

## Acknowledgments

This research was supported by the Intramural Research Program of the NIH,
National Institute of Allergy and Infectious Diseases.

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