The boundary of the outer disk D(R) is assumed reflecting, except for a small absorbing ... The Brownian motion in Ω with traps can be described by the Kramers- ...
Dwell time of a Brownian molecule in a microdomain with traps and a small hole on the boundary Adi Taflia1 and David Holcman1, 2 1
Department of Mathematics, Weizmann Institute of Science, Rehovot 76100, Israel. 2
D´epartement de Math´ematiques et de Biologie,
Ecole Normale Sup´erieure, 46 rue d’Ulm 75005 Paris, France
∗
Abstract We calculate the mean time a Brownian particle spends in a domain with traps and the number of bonds it makes before escaping through a small hole in the boundary. This mean time, called the Dwell time, depends on the backward binding rate (with the trap, e.g., scaffolding molecules), the mean time to reach the trap (forward binding rate), and the size of the hole. We estimate the mean and variance of the number of bonds made prior to exit. In a biochemical context, a quantitative signal occurs when the mean number of bonds exceeds a certain threshold, which may initiate a cascade of chemical reactions that have physiological consequences. We apply the present results to obtain estimates on the mean time a Brownian receptor spends inside a synaptic domain, when it moves freely by lateral diffusion on the membrane of a neuron and interacts at a synapse with scaffolding molecules.
∗
D. H. is supported by the program “Chaire d’Excellence”. This research is supported by the grant Human Frontier Science Program 0007/2006-C.
1
INTRODUCTION
Biochemical reactions may involve a small number of molecules, which diffuse inside cellular microdomains and can bind and unbind to and from agonist molecules (traps). The statistics 1 of the time a Brownian particle spends inside a microdomain and of the number of bonds it makes is of prime physiological significance. Many authors [1–5], to quote but a few, developed the diffusion theory of chemical reactions at a molecular level. They computed, among others, various rate constants using classical diffusion theory of diffusion with interactions with binding sites. Recently, chemical reactions in microdomains were described in terms of averaged equations [6]. Chemical reactions in closed microdomains were studied in [7], where some estimates on the mean and variance of the number of bound molecules in steady state were given. We consider here a microdomain Ω defined as a bounded domain, where a large fraction of the boundary ∂Ωr is impermeable to diffusing molecules, except for a small part ∂Ωa , through which molecules can enter or/and exit. The parts ∂Ωr and ∂Ωa of the boundary are described mathematically as reflecting and absorbing for the Brownian motion, respectively. The diffusing molecule can be trapped several times prior to exit through ∂Ωa . In the context of cellular biology and physiology, if the number of trapped molecules (the number of chemical bonds formed) exceeds a given threshold, a cascade of chemical reactions can be initiated, which ultimately affects the properties of the cell and/or its function. Our purpose is to estimate the number of chemical bonds a Brownian particle makes inside Ω before it exists. This number depends on the geometry of Ω and on the distribution of the traps it contains. More specifically, we want to approximate the mean time (dwell time) a Brownian particle spends inside the microdomain, including binding time, before it exits through ∂Ωa . For that purpose, we derive an asymptotic formula for the dwell time, the mean and variance of the number of bonds made before exit, when the ratio ε=
|∂Ωa | 2|∂Ω|
(1)
is small. An explicit formula for the dwell time was given in [8, 9] for domains Ω that do not contain traps. More specifically, if D denotes the diffusion constant, |Ω| the volume of the domain Ω and 2ε is the ratio of the absorbing to the total boundary, then for ε small, 2
the leading order term of the mean escape time τ (x) (for a molecule starting at position x, far from the entrance) is given by τ (x) =
|Ω| 1 log( ) + O(1). πD ε
(2)
In the first approximation the mean time τ (x) does not depend on the initial position x and is abbreviated as τ . In this article, we generalize (2) for domains containing an immobile trap. Each time the Brownian particle is trapped reversibly it stays trapped for a random time, whose average is the reciprocal of the backward binding rate k−1 . Upon leaving the trap the Brownian motion is resumed at a random point in the domain. Any number of trappings may occur prior to exit through ∂Ωa . We obtain first an explicit asymptotic approximation of the dwell time E(τD ) as a function of hτ i−, the mean conditional time to exit before binding, hT i− the mean time between consecutive trappings, m− the mean probability to exit before binding and the backward binding rate k−1 . The first result of this work is the general formula for the dwell time: 1−m E(τD ) = hτ i + m
Ã
!
1 + hT i . k−1
(3)
Second, we refine the asymptotic expression for E(τD ) to include the geometrical properties of Ω, the size of the small openings ε and and the radius of the trap site δ. We recall that the backward binding rate k−1 depends only on the local interactions between the Brownian ∆E
particle and the trap, as expressed in the Arrhenius law k−1 = Ce− kTe , where C is a constant that depends on the temperature Te , the electrostatic potential barrier ∆E generated by the binding molecule and the friction coefficient [10]. The paper is organized as follows: In the first part of the paper, we derive equation (3) by counting the number of times the Brownian particle is trapped prior to exit through ∂Ωa . In the second part, we estimate hT i, hτ i and m as a function of δ, ε. Although the present computations are carried out in two dimensions, they can be extended to three dimensions by using the techniques developed in [12]. Finally in the last part, we apply the present computations to study the interaction of receptors with scaffolding molecules (traps) inside the postsynaptic density (PSD), which is a specialized microdomain. We estimate there the number of bound receptors when a steady state flux is maintained at the absorbing boundary, assuming exiting receptors are lost. The role of the flux is to fix the number of free binding sites. It is not clear at present what mechanisms regulate the number and type 3
of receptors in the PSD. The number of receptors defines and controls the synaptic weight; therefore any fluctuation results in a variation in the weight and thus affects the reliability of the synaptic transmission.The regulation of synaptic plasticity is a fundamental process underlying learning and memory [13, 14] and recently, single molecule tracking has revealed that the number of postsynaptic receptors, which participate in the synaptic transmission, is not fixed but it changes due to constant trafficking of receptors on the membrane of neurons. Observations show that receptors move in and out of synaptic regions [15–17] and consequently many questions have been raised: in particular, what determines the time a receptor spends inside a synapse? How receptors are stabilized inside a synapse? How long they stay inside synaptic microdomains? Such questions are partially answered in the present paper.
Model of the molecular dynamics in a domain with traps
A Brownian particle diffuses in a circular annulus Ω, of outer radius R and inner radius δ. The boundary of the outer disk D(R) is assumed reflecting, except for a small absorbing arc ∂Ωa . The boundary of the inner disk D(δ) is assumed absorbing. A Brownian particle absorbed in ∂Ωa is assumed never to return to Ω, while a particle absorbed in the boundary of the inner disk reappears at an exponentially distributed time with rate k1 at a uniformly distributed point in and resumes its Brownian motion. We denote the reflecting part of the boundary ∂Ωr = ∂D(R) − ∂Ωa . The inner disk D(δ) represents a coarse-grained effective model of small-scattered traps (e.g., scaffolding molecules), which are pulled together to form a single bigger trap, while preserving the absorption flux and the distribution of the trapping time. This simplified model is presumably derivable by homogenization of a model of diffusion in a potential landscape with many small, but deep wells (relative to the thermal energy), which represent the potential of the chemical bond in the traps (Fig. 1). The Brownian motion in Ω with traps can be described by the Kramers-Smoluchowski stochastic differential equation ∇V (X) X˙ + = γ
s
2εe w, ˙ γ
(4)
where X is the position at time t, V (X) is the potential of the trap, γ is the dynamical viscosity (friction) coefficient, εe =
kTe m
(kB is Boltzmann’s constant, Te is the temperature, 4
and m is the reduced mass of the particle), and w˙ is a δ-correlated standard Gaussian white noise. A sufficiently deep potential well traps the Brownian motion for exponentially distributed random times with rate k−1 , given by Kramers’ formula for the Arrhenius rate. The probability density function (PDF) p to find X at time t in the surface element x + dx satisfies the Fokker-Planck Equation (FPE)
∂p(x, t) = D∆p(x, t) − ∇ · (∇V (x)p(x, t)) for x ∈ Ω ∂t J (x, t) · n = 0 for x ∈ ∂Ωr
(5)
(6)
p(x, t) = 0 for x ∈ ∂Ωa ,
(7)
where n is the external normal at the boundary, the net flux J is given by J (x, t) = −D∇p(x, t) + ∇V (x)p(x, t). We denote by TxA the first time the molecule arrives at the absorbing boundary ∂Ωa , when it started at position x. The mean first passage of a particle starting at x, to ∂Ωa is given by [10] E(TxA )
= =
Z ∞ Z0 ∞
d P r{TxA < t}dt Zdt
t
0
Ω
p(y, t|x)dydt,
where p(y, t|x) is the PDF of the process X, conditioned on the initial position x, that is, as t goes to zero, p(y, t|x) → δ(x − y), where δ is the Delta-Dirac function.
Assumptions about the geometry of the microdomain
We estimate asymptotically the Dwell time E(τD ) of a molecule inside the annulus Ω = D(R)−D(δ). The case of a general domain is left open. By definition, E(τD ) is the mean time 5
to exit, averaged over an initial uniform distribution. We make the following assumptions: the ratio of the absorbing to the reflective boundary of D(R) is small (ε Tx satisfies the equation with mixed boundary conditions [20]:
∆q = 0 on Ω, ∂q(x) = 0 on ∂Ωr , ∂n q(x) = 0 on ∂Ωa ,
(14)
q(x) = 1 on ∂D(δ), where ∂Ωa is the small opening, ∂Ωr is the remaining part of the external boundary, which is reflecting. In polar coordinates (r, θ), the portion of the boundary ∂Ωa is parameterized by θ when |θ − π| ≤ ε. For an annulus Ω, we obtain an explicit representation of the solution by using methods, developed in [21–23]. By the method of separation of variables, the solution q(x) of equation (14) is given by q(r, θ) = We denote β =
δ . R
∞ h ³ ´ ³ δ ´n i a0 X r n r an + + bn cos(nθ) + γ log . 2 R r δ n=1
To estimate the coefficients an and bn we use the boundary conditions
on ∂D(δ) and on r = R, thus we get 10
∞ h ³ ´ i a0 X δ n + an + bn cos nθ = 1, 2 R n=1 ∞ X h an γ bn i n − β n cos(nθ) + = 0 for |θ − π| > ε R R R n=1 ∞ h i X 1 1+ an + bn β n cos(nθ) + γ log = 0 for |θ − π| ≤ ε. β n=1
(15) (16) (17)
From equation (15), we obtain the following identities: a0 = 2 bn = −an β n for n ≥ 1. Using the identities above and (16) and (17), we obtain the double series equations
P∞
n=1
1+
P∞
n=1
n
h
an R
an 2n β R
i
cos(nθ) +
γ R
for |θ − π| > ε
(18)
an − an β 2n cos(nθ) − γ log(β) = 0, for |θ − π| ≤ ε.
(19)
h
+
i
2β 2n 1−β 2n
Substituting cn = an (1 + β 2n ) and Hn = c0 2
+
γ+
P∞
cn n=1 1+Hn
P∞
n=1
= 0,
equations (18),(19) have the following form
cos(nθ) = 0, θ ∈ [π, π − ε]
(20)
ncn cos(nθ) = 0, θ ∈ [0, π − ε],
(21)
c0 = 1 − γ log(β). 2
(22)
where
The asymptotic solution of equations (20)-(21) uses the double series expansion, developed in [21, 22]. In Appendix A, we provide the details of the mathematical derivations. By using these results, We can now obtain the asymptotic expression for the leading coefficient c0 : h
c0 = −2γ 2 log
i 1 + 2 log 2 + 4β 2 + O(β 2 , ε) . ε
(23)
Using equation (22) and (23), we get the expression γ=−
log
1 β
1 . + 2 log + 2 log 2 + 4β 2 + O(β 2 , ε) 1 ε
11
(24)
We define now a new constant, which will play a crucial role later on, α :=
log
1 β
1 , + 2 log + 2 log 2 + 4β 2 + O(β 2 , ε) 1 ε
(25)
To remember that α depends on β and ε, we denote it α(β, ε). For ² fixed and δ small enough, the remaining coefficients an , bn does not really count, indeed using the expression of cn given in the appendix, we get that cn = O(α(β, ε)), and cn ∼ O(α(β, ε)) 1 + β 2n bn = −cn β n ∼ O(α(β, ε)β n ).
an =
These estimates show that for δ small, the leading order term of q is given by 1 − α(β, ε) log(r/δ) + O(β)
for r ∼ δ
qδ (r, θ) =
1 − α(β, ε) log(r/δ) + O(α) for r ∼ R.
(26)
Finally, because p(x) = 1 − q(x), averaging over a uniform distribution we obtain the asymptotic expression for the mean number of bound, Ã
!
1 1 1 Z + O(β 2 , ε) m= p(r, θ)rdrdθ ∼ α log − |Ω| Ω β 2 More generally, the computation of mρ can be extended for any radially symmetric distribution. For example when 1 2 2 π(R1 −δ )
for δ < r < R1 (27)
ρ(r, θ) =
0
for R1 < r < R,
then µ
( mρ =
log
1 β
1 2 1− δ 2
) log Rδ1 − 12 +
R 1
1 2
³
δ R1
´2 ¶
+ 2 log 1ε + 2 log 2 + 4β 2 + O(β 2 , ε)
12
.
(28)
Using the expressions for m and the formula (12) the mean number and the variance of bindings are given by 2 log 1ε + 2 ln 2 + 1 − mδ = mδ log β1 − 12
Mb =
³
2 log 1ε + 2 ln 2 +
Vb =
1 2
´³
1 2
+ o(1)
log β1 + 2 log 1ε + 2 log 2
(29) ´
(log β1 − 12 )2
.
These expressions are valid for ε ¿ 1 fixed, but are uniform in β for β
