proof copy [el10671] 170003pre - ENS Biologie

PROOF COPY [EL10671] 170003PRE PHYSICAL REVIEW E 81, 1 共2010兲 1

Threshold activation for stochastic chemical reactions in microdomains

2 3 4 5

Département de Mathématiques et de Biologie, Ecole Normale Supérieure, 46 rue d’Ulm, 75005 Paris, France and Department of Applied Mathematics, Tel-Aviv University, Tel-Aviv 69978, Israel 共Received 20 November 2009兲 The mean time to reach a threshold 共MTT兲 is the mean first passage time for the number of bound molecules to reach a given value. In the theory of chemical reactions involving a small number of ligands and molecules, the MTT represents the first time a given number of binding sites is formed. In that context, the MTT can be used to characterize the stability of chemical processes, especially when they underlie a biological function. Using a Markov-chain description, we compute here the MTT, in terms of fundamental parameters, such as the number of molecules, the ligands and the forward and backward binding rates. We find that the MTT depends non-linearly on the threshold T, and this result may have several applications, ranging from cellular biology to synaptic plasticity. We confirm our analytical computations with Brownian simulations.

F

14

OO PR

6 7 8 9 10 11 12 13

K. Dao Duc and D. Holcman

DOI: XXXX

16

␭

I. INTRODUCTION

M + S MS, k−1

where ␭ is the forward rate at which a M-molecule encounters one of the free targets and k−1 is the backward binding rate at which M − S molecules dissociate. The MTT ␶T for the amount of MS molecules at time t, to reach a threshold T is the expectation of the random time

␶T = inf兵t ⬎ 0兩MS兩共t兲 = T其,

L1

[E

00

70

]1

71

␮

M→ 쏗 .

共3兲 77

For that case, we introduce a two-dimensional Markov chain and by counting the number of paths in a Markov graph, we obtain a general expression 共40兲 for the probability and the meant time to reach the threshold T 共49兲. When the number of binding molecules is large compared to the threshold T, we obtain some asymptotic expressions, confirmed by Brownian simulations. Finally, when there are two competitive molecular pathways, we shall evaluate the probability and the MTT for one of them to be activated before the other. 1-1

63 64 65 66 67 68 69 70 71 72 73 74 75

共2兲 76

M + S→ MS,

RE

PROOF COPY [EL10671] 170003PRE

where 兩MS兩共t兲 is the number of MS molecules at time t. We estimate here ␶T in several cases: for an ensemble of the targets initially free and distributed on the surface of a closed microdomain, we compute the MTT analytically when the backward rate vanishes 共k−1 = 0兲 and later on when k−1 ⬎ 0. We further confirm these formulas with Brownian simulations. Our main results are summarized by formulas 共17兲–共20兲, 共22兲, and 共23兲 which extend previous work on the first passage time for discreet process 共ch. XII, p. 292 关11兴 or on diffusion controlled readtions p. 272 关12兴. We shall further extend our analysis to diffusing molecules that can disappear 共with a Poissonian rate兲 before binding the target sites, modeled as ␭

57 58 59 60 61

共1兲 62

06

1539-3755/2010/81共4兲/1共12兲

56

3P

Molecular activation occurring in cellular microdomains depends on the binding-unbinding of ligands to specific targets 关1,2兴. This is for example the case at synapses, where synaptic plasticity, a long lasting process underlying learning and memory 关3兴 can be induced when the concentration of calcium reaches a certain threshold 关4兴. Another example concerns the cellular response to a double strand DNA 共dsDNA兲 break: the cell can “sense” the number of breaks and may decide to undergo apoptosis or not. Interestingly, a single dsDNA break can be detected and this event is sufficient to activate a global cellular response. The concept of reaching a threshold as the starting point of a cellular response is ubiquitous in biology. We shall mention two other fundamental examples: in the patterning process, occurring in the embryo development, cell differentiation is controlled by a gradient concentration of morphogens and interestingly, the cell fate can change by a small difference in that concentration 共the concentration of the decapentaplegic gene 共DPP兲 in insects can activate different genes at different thresholds 关5兴. Another example concerns the first step of cellular division, where chromosomes need to be attached before being separated 关6,7兴 We present here several scenarios where an ensemble of particles 共molecules, proteins, ions, ligands兲 interacts with immobile targets via binding and unbinding. We compute the mean first time MTT that the number of bound targets reaches a threshold value T, which is a key event to induce a cellular response. However, because studying the MTT involves discreet events, we cannot use the standard equationsof mass-action law describing chemical reactions. Recently several efforts were made 关8,9兴 to model chemical reactions in small domains. In 关10兴, using a Markov chain description, we developed a theory of chemical reactions in microdomains based on the dynamis of few Brownian particles. Although we obtained estimates for the number and variance of bound molecules, we did not consider the mean time to reach a threshold, which we are now studying. We consider M Brownian molecules inside a microdomain that can bind to immobile targets S modeled generically as

PY

17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55

CO

15

PACS number共s兲: 05.40.Ca, 87.10.Mn, 82.20.Uv, 87.18.Tt

©2010 The American Physical Society

78 79 80 81 82 83 84 85 86

PROOF COPY [EL10671] 170003PRE PHYSICAL REVIEW E 81, 1 共2010兲

K. DAO DUC AND D. HOLCMAN 87 88 89

We conclude with some applications to the spindle checkpoint, an event that precedes the mitotic phase in cell division.

90

II. MTT FOR STOCHASTIC CHEMICAL REACTIONS

91 92 93 94 95 96 97 98 99 100

We now use a Markov chain to compute the mean first passage time for the number of bound molecules to reach the threshold T. The substrate S and the diffusing molecules M interact as described by equation I. To derive a markov equation for the probability density functions pk共t兲 = Prob兵兩MS兩共t兲 = k 兩兩MS兩共0兲 = 0其 that the number of bound molecules 兩MS兩 at time t is equal to k, we study between time t and t + ⌬t, the transitions to the k state, coming from states k − 1, k, and k + 1 with transition rates ␭k−1 , −共␭k + k−1k兲 , k−1共k + 1兲关10兴. pk 共for 0 ⱕ k ⱕ S0兲 satisfy

101

p˙0 = − ␭0 p0 + k−1 p1 ,

102

p˙k = − 共␭k + k−1k兲pk + ␭k−1 pk−1 + k−1共k + 1兲pk+1

=

冕

+⬁

关1 − pT共t兲兴dt.

共10兲 127

0

Equivalently using the normalization condition

128

T

OO PR

PY

p˙S0 = − 共␭S0 + k−1S0兲pS0 + ␭S0−1 pS0−1 ,

104

␭k = ␭共S0 − k兲共M 0 − k兲,

107 108 109 110 111 112 113 114 115

which is the rate for one of the M 0 − k free molecules to reach the S0 − k free binding sites. ␭ is the binding rate for one molecule M to a single target, it is the reciprocal of the mean first passage time ¯␶ for a particle to a target. When the target is small enough, ¯␶ can be approximated by the small hole formulas 关13–16兴. To estimate the mean first time that the number of bound molecules reaches the threshold T, we impose in Eq. 共4兲 that the state 兩MS兩 = T is absorbing, which leads to the modified system:

␭T−1

1 ak+1 + 共k + 1兲k−1 ␭k ␭k

共14兲

,

for

0 ⱕ k ⱕ T − 2.

共15兲

142

143

70

00

1 1 1 + + ... + , ␭0 ␭1 ␭T−1

共16兲

146

T−1

1 1 . 兺 ␭ k=0 共M 0 − k兲共S0 − k兲

3P

共7兲

共17兲 147

共8兲

␶Tirrev ⬇

RE

In particular, when M 0 = S0 and M 0 Ⰷ 1, the asymptotic for- 148 mula for Eq. 共17兲 becomes 149

The MTT ␶T is given by

T . ␭M 0共M 0 − T兲

共18兲

150

In addition, when the diffusing molecules largely exceed the 151 number of targets 共M 0 Ⰷ S0 , T兲, we further obtain from Eq. 152 共17兲, the asymptotic formulas, 153

+⬁

共9兲

0

1-2 PROOF COPY [EL10671] 170003PRE

1

aT−1 =

=

where ␶T is the first hitting time to the threshold T,

Prob兵␶T ⬎ t其dt

141

共6兲

pT共t兲 = Prob兵␶T ⱕ t其,

冕

共13兲 140

equivalently

␶Tirrev =

122

¯␶T =

139

When the binding is irreversible 共k−1 = 0兲, the MTT ¯␶T is the 144 sum of the forward rates: 145

where pk共t兲 = Prob兵兩MS共t兲兩 = k其 with 0 ⱕ k ⱕ T. By definition

␶T = inf兵t,兩MS兩共t兲 = T其.

138

1 = ␭T−1aT−1 .

ak =

121

126

0 ⬍ k ⬍ T − 1,

]1

p˙T = ␭T−1 pT−1 .

137

0 = − 关␭T−1 + k−1共T − 1兲兴aT−1 + ␭T−2aT−2 ,

共5兲

0 ⬍ k ⬍ T − 1,

120

125

136

71

for

p˙T−1 = − 关␭T−1 + k−1共T − 1兲兴pT−1 + ␭T−2 pT−2 ,

124

− 1 = − ␭0a0 + k−1a1 ,

for

p˙k = − 共␭k + k−1k兲pk + ␭k−1 pk−1 + k−1共k + 1兲pk+1

119

123

132 133 134 135

06

118

where ak = 兰+⬁ 0 pk共t兲dt. To obtain an analytical expression for ¯␶T, we integrate Eq. 共6兲 between 0 and +⬁ with the initial conditions p0 = 1 , pk = 0. Using that pT共+⬁兲 = 1 and pk共+⬁兲 = 0 for k ⬍ T, we get

L1

117

131

0 = − 共␭k + k−1k兲ak + ␭k−1ak−1 + k−1共k + 1兲ak+1

p˙0 = − ␭0 p0 + k−1 p1 ,

116

共12兲

0

共4兲

106

130

¯␶T = 兺 ak ,

[E

where

129

T−1

CO

0 ⬍ k ⬍ S0 ,

共11兲

we have the general expression

F

for

103

105

兺0 pk共t兲 = 1,


THRESHOLD ACTIVATION FOR STOCHASTIC CHEMICAL… 1

T−1

T−1

0.8

k=0

0.6

T−1

0.4

3 + k−1 兺

k共k − 1兲共k − 2兲 + ... ␭k␭k−1␭k−2␭k−3

T−1

T−1

0.2

OO PR

冉

k=3

j = 兺 k−1 兺

0

j=0

−0.2

T−1

−0.4

=

−0.6

T−1

1 k共k − 1兲 k 2 + k−1 兺 + k−1 兺 ␭k k=1 ␭k␭k−1 k=2 ␭k␭k−1␭k−2

¯␶T = 兺

k=j

k! k 共k − j兲!兿i=k−j ␭i

冉冊兺

1 k−1 兺 ␭ j=0 ␭

j T−1 k=j

160

161

冊

162

k! 共M 0 − k − 1兲! 共S0 − k − 1兲! . 共k − j兲! 共M 0 − k + j兲! 共S0 − k + j兲!

163

−0.8

−0.5

35

0

0.5

1

T−1

30

25

15

5

6

7

8

9

10

11

T

when

T k−1 T − log 1 + 共␭S0兲2 M 0 − T M0 − T

when

S0 Ⰷ M 0,T,

␶Tirrev

冉冊

k−1 T + 2 2␭ M 20

3

,

when

S0 = M 0 Ⰷ T.

168 169

冧

共21兲 170

Using the expression for ␶irrev 关Eq. 共19兲兴, we finally obtain 171 the asymptotic expressions: 172 ¯␶

共19兲 ⬇

M 0,S0 Ⰷ T.

In Fig. 1, we plot the MTT ␶Tirrev for several values of the threshold T and we compare it with Brownian simulations performed in a circular disk ⍀ = D共R兲, which boundary is reflecting except at the targets. When k−1 ⬎ 0, the analytical expression for ¯␶T is given by

冦

再冉冊冋冉冊册冎冉冊册冎再冉冊冋

173

1 T S0 T k−1 − log log 1 + + ␭M 0 S0 − T ␭M 0 S0 − T S0 − T when

M 0 Ⰷ S0,T,

1 M0 T M0 k−1 log + − log ␭S0 M0 − T ␭S0 M 0 − T M0 − T when

冋

S0 Ⰷ M 0,T,

冉冊册

T k−1 T 2 1+ 2␭2 M 20 ␭M 0

2

when

S0 = M 0 Ⰷ T.

冧

共22兲 174

Furthermore, when T Ⰶ S0 and T Ⰶ M 0 respectively, in the 175 two first regimes 关Eq. 共22兲兴, we obtain the refined estimates 176 1-3


␶Tirrev +

RE

T ␭M 0S0

冊册

M 0 Ⰷ S0,T,

3P

S0 Ⰷ M 0,T,

冉

when

166 167

,

00

155 156 157 158 159

冧

1 M0 when log ␭S0 M 0 − T

冊册

70

154

冦

M 0 Ⰷ S0,T,

冉

T T k−1 − log 1 + 2 共␭M 0兲 S0 − T S0 − T

]1

␶Tirrev ⬇

1 S0 when log ␭M 0 S0 − T

冦

冋冋

␶Tirrev +

71

FIG. 1. 共Color online兲 MTT: Up, we present the trajectories of diffusing molecules in a microdomain containing five binding sites on the boundary. Down: we plot the time ␶Tirrev as a function of the threshold T. We present the Brownian simulations 共dash line, variance in black兲, the theoretical formula 共17兲共dash-dot line兲 and its approximation 共19兲共solid line兲 for a circular disk in the irreversible case 共k−1 = 0兲. The other parameters are S0 = 15, M 0 = 10, ⑀ = 0.05, D = 0.1 ␮m2 s−1, and the radius of the disk R = 1 ␮m 共we run 200 simulations兲.

¯␶T ⬇

06

4

L1

5

(b)

k=j

k! 共M 0 − k − 1兲! 共S0 − k − 1兲! . 共k − j兲! 共M 0 − k + j兲! 共S0 − k + j兲!

This sum can be further approximated for the three following regimes M 0 Ⰷ S0 , T , S0 Ⰷ M 0 , T, and S0 = M 0 Ⰷ T, by using the k 1 first-order expansion in ␭−1 only and the sum 兺N1 k = log共N兲 + O共1兲. We obtain,

[E

10

0

j T−1

共20兲 165

PY

20

164

冉冊兺

1 k ¯␶T = 兺 −1 ␭ j=0 ␭

CO

Mean first time to T

(a)

Finally, we obtain

F

−1 −1


K. DAO DUC AND D. HOLCMAN

¯␶ ⬇

when

when

冊冉冊册 T S0

M 0 Ⰷ S0 Ⰷ T,

1 T k−1 1+ 1+ 2 ␭M 0S0 ␭S0

OO PR

177

冦

冋冉冋冉

1 T k−1 1+ 1+ 2 ␭M 0S0 ␭M 0

冊冉冊册 T M0

S0 Ⰷ M 0 Ⰷ T.

冧

,

共23兲

,

We conclude that for ¯␶ Ⰷ 1, the time to threshold T ␶T varies quadratically with the narrow escape time ¯␶ 共as ¯␶ = ␭1 兲, however, it is a nonlinear increasing function of T. These computations we presented are quite general and can be applied to describe the mean time to a given number of bound molecules for any chemical reactions. In particular, changing the threshold can be seen as a modulatory mechanism.

185 186

III. MTT FOR CHEMICAL REACTIONS IN A DOMAIN WITH KILLING PROCESS

187 188 189 190 191 192 193 194 195 196 197

We now study in a microdomain with binding sites, the MTT of diffusing molecules which can also be killed before hitting the sites. In that case, the threshold may never be reached. We shall compute first the probability to the threshold and second the MTT. We treat the case of irreversible binding sites only and model the killing process as Poissonian of parameter ␮. To derive the probability to reach the threshold T, we use a two-dimensional Markov chain for the joint probability density function pk,m共t兲 that at time t, there are k bound molecules and it remains m − k free diffusing molecules.

198

pk,m共t兲 = P兵兩MS兩共t兲 = k,w共t兲 = m,兩MS兩共0兲 = 0,w共0兲 = M 0其,

199 200 201 202 203 204 205 206 207 208 209

where w共t兲 = 兩M兩共t兲 + 兩MS兩共t兲. In the irreversible case, the transitions to the state 共k , m兲 between time t and t + ⌬t can only occur from the states 共k − 1 , m兲, 共k , m兲 and 共k , m + 1兲. To compute the probability that the threshold T is achieved, we study the chain 共k , m兲 of having k bound molecules and m-k free remaining molecules. By imposing a boundary condition at 共T , m兲 and 共k , T − 1兲, we will obtain the probability that the threshold is reached before the molecules are degraded by summing over the state 共T , m兲 for m = T . . . M 0. We represent the transition diagram between states in Fig. 2 and the master equations are

210

p˙0,M 0 = − 共␭M 0S0 + ␮ M 0兲p0,M 0 ,

]1 冉

1 ␮ m共␭S0 + ␮兲 ␭S0 + ␮

共S0 − k兲!共M 0 − k兲兿kj=0关␮ + ␭共S0 − j兲兴 0 ⬍ k ⬍ T,

for

T ⱕ m ⱕ M0 ,

pk,T−1共⬁兲 = ␮共T − k兲qk,T

0 ⱕ k ⱕ T − 1, 1-4


共27兲 230

pT,m共⬁兲 = ␭共S0 − T + 1兲共m − T + 1兲qT−1,m

T ⱕ m ⱕ M0 ,

229

T − 1 ⬍ m ⬍ M0 ,

RE

for

for

3P

p˙k,T−1 = ␮共T − k兲pk,T

M 0−m

共26兲

225 226 227 228

␭ kS 0!

for

for

冊

00

p˙T,m = ␭共S0 − T + 1兲共m − T + 1兲pT−1,m

224

1 , M 0共␭S0 + ␮兲

70

q0,m =

0 ⬍ k ⬍ T,

223

共25兲

pk,m = 1.

71

216

兺

k=0,m=0

qk,M 0 =

for

220 221 222

M 0,N0

q0,M 0 =

p˙k,M 0 = ␭共S0 − k + 1兲共M 0 − k + 1兲pk−1,M 0 − 关␭共S0 − k兲 + ␮兴共M 0 − k兲pk,M 0

共24兲 219

To derive the steady state probabilities pT,m共⬁兲 and pk,T−1共⬁兲, we shall now use that the only absorbing states 共T , m兲 and 共k , T − 1兲 are given for T ⱕ m ⱕ M 0 and 0 ⱕ k ⱕ T − 1 共see Fig. 2兲. Thus by integrating over time system 共24兲, we obtain that

T − 1 ⬍ m ⬍ M0 ,

217 218

where ␭ and ␮ are, respectively, the forward binding rate and the killing rate. The initial condition reads p0,M 0共0兲 = 1 and the normalization condition is

p˙0,m = ␮共m + 1兲p0,m+1 − 共␭S0m + ␮m兲p0,m for

T−1⬍m

for

0 ⬍ k ⬍ T,

06

215

⬍ M 0,

L1

214

− 关␭共S0 − k兲 + ␮兴共m − k兲pk,m

[E

213

p˙k,m = ␭共S0 − k + 1兲共m − k + 1兲pk−1,m + ␮共m + 1 − k兲pk,m+1

PY

212

CO

211

FIG. 2. Diagram of transition between states. Transition rates 共1兲: ␮共m − k + 1兲共2兲: ␭共S − k + 1兲共m − k + 1兲共3兲: ␭共S − k兲共m − k兲共4兲: ␮共m − k兲 where ␭ and ␮ are, respectively, the forward binding rate and the killing rate.

F

179 180 181 182 183 184

178

for

0 ⱕ k ⱕ T − 1,

231

共28兲 232 233

共29兲 234 共30兲 235



关␮qk,m+1 + ␭共S0 − k + 1兲共m − k + 1兲qk−1,m兴 qk,m = 236 共m − k兲关␮ + ␭共S0 − k兲兴 0 ⬍ k ⬍ T,

T − 1 ⬍ m ⬍ M 0,

for

共31兲

237

where for 0 ⱕ k ⱕ T − 1, T ⱕ m ⱕ M 0,

OO PR

238

qk,m =

239

240

冕

⬁

pk,m共t兲dt.

共32兲

0

A. Probability to reach the threshold

The probability PT to reach the threshold is equal to the probability to reach any of the states 共T , m兲 for m = T , .. M 0, that is

F

CO

241 242 243

Probability to reach T

⫻ 共m + 1 − k兲

m/l = 1 0.9

0.8

0.7

m/l = 5

0.6

0.5

0.4

0.3

m/l = 10 0.2

0.1

M0

PT = 244

共33兲

␮

兺 0ⱕi ,..,i 1

M 0−m

兿 ⱕk i ␭共S0 − i j兲 + ␮ j

关␭共S0 − k兲 + ␮兴共m − k兲

.

We proceed with the computation of the probability PT to reach the threshold. We get from Eq. 共30兲, for T ⱕ m ⬍ M 0 that

252

pT,m共⬁兲 = ␭共S0 − T + 1兲共m − T + 1兲qT−1,m ,

i=0

兺 0ⱕi ,..,i 1

254

M 0−m

共35兲

冊 ␮

冊

for

冉兿 i=0

冊

␭共S − i兲 . ␭共S − i兲 + ␮

共38兲

Finally, using that the probability is given by 1-5


1

1

1.5

2

2.5

3

3.5

4

4.5

T

RE

259

pT,M 0共⬁兲 =

1.5

FIG. 3. 共Color online兲 Probability and MTT in a microdomain where ligands can be killed before binding. Up: we plot the probability PT to reach the threshold as a function of T, for dif␮ ferent values of the ratio ␭ 共1,5,10兲. The exact formula 共40兲共solid line兲 is compared with the approximation 共43兲共dash line兲. We also compare with Brownian simulations of M molecules 共dash-dot line兲, with diffusion coefficient D moving inside a circular disk of radius R. The binding sites are of size ⑀. Down: we plot the conditional MTT as function of T for ⑀ = 0.05. The mean first passage time 共the initial position is at the center兲 to a binding site of size ⑀ 1 R2 4␲ 1 is approximated by ␭ = ␶ = 共 D 兲关log共 ⑀ 兲 + 4 兴共see 关15兴兲. The other parameters are R = 1 ␮m, D = 1 ␮m s−2, S = 5, and M = 10. Number of Brownian simulations= 250.

共37兲

T−1

258

(b)

T − 1 ⬍ m ⬍ M0 ,

256

we also have for m = M 0,

2

3P

冉

1 ␮ m共␭S0 + ␮兲 ␭S0 + ␮

5.5

2.5

0 0.5

Using Eq. 共28兲, q0,m =

257

共36兲

k

M 0−m

5

3

0.5

. 兿 ⱕT−1 i ␭共S0 − ik兲 + ␮

5

3.5

00

⫻

4.5

4

70

= 253

␭共S0 − i兲 ␭共S0 − i兲 + ␮

4

]1

冉兿

T−1

3.5

71

249 250 251

3

06

共34兲

248

2.5

T

Conditonnal mean time to T

册

2

4.5

L1

i=0

qk,m =

␭共S0 − i兲 ␭共S0 − i兲 + ␮

1.5

5

[E

冋兿

1

(a)

To estimate PT, we study the system of Eqs. 共26兲–共31兲 and for 0 ⬍ k ⬍ T and T ⱕ m ⬍ M, we have derived in the Appendix A k−1

255

0

PY

245 246 247

兺 pT,m共⬁兲.

m=T



␭S noulli distribution of parameter ␭S+ ␮ , and the probability of the binding number is a binomial distribution of parameters ␭S 共M , ␭S+ ␮ 兲. Finally, the probability to reach the threshold is equivalent to have at least T bounds, and thus we obtain formula 共43兲. Interestingly, as shown in Fig. 3, already for r = 10, the exact formula cannot be well approximated by the approximation 共41兲 and the analytical solution should be used. The probability is a decreasing, inverse sigmoid type function as function of the threshold T.

M0

PT = 260 261

冉冊

␮ ,T = PT = ␭

␭共S − i兲 ␭共S − i兲 + ␮

冉

i=0

⫻ 1+

263

+

␮

1

␮

兺 0ⱕi ⱕ..ⱕi

M 0−T

兿 ⱕT−1 i ␭共S − ik兲 + ␮ k

冊

␮ ␭S ,T ⬇ ␭ ␭S + ␮

T

1+

冊

␮ 兩G1兩 ␭S + ␮

冉

i ␮ ␮ 兩Gi兩 . . . + ␭S + ␮ ␭S + ␮

冊

冊

M−T

PY

冉

冊冉

CO

冉冊冉

兩G M 0−T兩 , 共41兲

k

k=0

冊冉

␮ ␭S + ␮

冊

,

冉冊兺冉冊冉 M ␮ ,T ⬇ ␭ k=T k

␭S ␭S + ␮

冊冉

␮ ␭S + ␮

冊

=

共43兲

.

E共␶K,兩MS兩 = k,兩M兩 = m − k兲 =

共44兲

1 . ␭k,m + ␮共m − k兲

共45兲

70

n

i j−1

兺兺 j=1 k=i

j−1

1 关␭共S − k兲 + ␮兴共M − j + 1 − k兲

i j−1⫽i j

otherwise. 1-6

冧

␴ = 共0,T兲

,

RE

1

+ 兺 j=1 关␭共S − i j兲 + ␮兴共M − j + 1 − i j兲

if

3P

00

冦

1

兺 k=0 关␭共S − k兲 + ␮兴共M − k兲

n−1

301

302

]1

¯␶共␴兲 =

T−1

299

303

The random times along the path ␴ are independent, thus the 304 total mean time ¯␶共␴兲 is the sum of all the mean times 305

306 307


1 . ␭k,m + ␮共m − k兲

298

300

Similarly, the mean time to killing is

M−k

This formula has the following interpretation: when there are many binding sites compared to the number of diffusing molecules, the binding events become independent. Consequently, the probability to bind can be approximated by Ber-

308

1 ␭k,m ␭k,m ␭k,m + ␮共m − k兲

71

p

k

E共␶ MS,兩MS兩 = k,兩M兩 = m − k兲 =

06

which can written as 共see Appendix B兲 M

275 276 277 278

289 290 291 292 293 294 295 296 297

k

共42兲

272

274

To get the conditional MTT, we first compute the mean time ¯␶共␴兲 for a trajectory parametrized by ␴ = 共i0 , i1 , i2 , . . . , in兲 where 0 = i0 ⱕ i1 ⱕ . . . in−1 ⬍ in = T and 1 ⱕ n ⱕ M − T + 1, that follows a path in the Markov diagram 共see Fig. 6 in Appendix B兲. When there are k bound molecules and it remains m − k free molecules, we shall estimate the mean transition time from this state 共k , m兲, to the state 共k + 1 , m兲. This event is Poissonian with rate ␭k,m = ␭共S − k兲共m − k兲 and the probability ␭ of binding before a molecule is killed is given by ␭k,m+␮k,m共m−k兲 . Thus, the conditional mean binding time is

L1

冊兺冉

T−1+k

281 282 283 284 285 286 287

288

[E

冉冊冉

T M−T

279 280

B. Conditional MTT

where 兩Gk兩 is the number of k tuples 共i1 , ..ik兲 where 0 ⱕ i1 ⱕ . . . ⱕ ik ⱕ T − 1. Using that 兩Gk兩 = 共 T−1+k k 兲, we obtain

␮ ␭S p ,T ⬇ ␭ ␭S + ␮

273

. 共40兲

F

p

269

271

冊

and T,

We show in Fig. 3, the graph of PT as a function of the threshold T for various values of the ratio r = ␮␭ . When TS Ⰶ 1, we obtain the approximation

+ ...

270

␮ ␭

+ ... 兺 0ⱕi ⱕT−1 ␭共S − i1兲 + ␮

1

264

268

冉兿

T−1

OO PR

262

266 267

共39兲

we finally obtain, introducing the variable p

265

兺 pT,m共⬁兲, m=T

共46兲



The probability P共␴兲 that the dynamics follows the path ␴ is

冦冉

T−1

␭共S − i兲

兿 i=0 ␭共S − i兲 + ␮

P共␴兲 =

T−1

␭共S − i兲 ␭共S − i兲 + ␮

OO PR

兿 i=0

␴ = 共0,T兲

if

冊冉兿

n−1 k=1

␮ ␭共S − ik兲 + ␮

otherwise.

310

冧

冊

IV. COMPETING THRESHOLDS

k1

M + S1→ MS1 ,

共47兲

Finally, for a forward binding constant ␭, a killing rate ␮, the conditional MTT ¯␶T共␭ , ␮ , T兲 is

313

¯␶T共␭, ␮,T兲 = 兺 ␶共␴兲P共␴兩T is reached兲,

314

CO

p

k2

共48兲

冉冊

共49兲

,

␮ ,T ␭

PY

where p共 ␮␭ , T兲 is the probability to reach the threshold computed in the previous subsection 关formula 共40兲兴. We now ␮ 317 approximate ␶T共␭ , ␮ , T兲 at first order in ␭ , which means that 318 we neglect all the paths ␴ = 共i0 , i1 , i2 , . . . , in兲 such that n ⬎ 2, ␮ 319 关the probability for the other paths is at least of order 共 ␭ 兲2兴. 320 Consequently, considering ␴ = 共0 , T兲 and ␴ = 共0 , i , T兲 with 0 ␮ 321 ⱕ i ⱕ T − 1. At first order, using formula 共41兲, p共 ␭ , T兲 = 1 ␮ 322 + o共 ␭ 兲. In addition, when M Ⰷ T, we obtain from relations 323 共46兲 and 共47兲 the approximations for 0 ⱕ i ⱕ T − 1 315 316

349 350 351 352 353 354 355 356 357

p˙0,0 = − 共k1ms1 + k2ms2兲p0,0 ,

358

p˙i,0 = − 共k1共s1 − i兲 + k2s2兲共m − i兲pi,0 + k1共s1 − i + 1兲共m − i + 1兲pi−1,0 , p˙0,j = − 关k1s1 + k2共s2 − j兲兴共m − j兲p0,j

L1

册兺

k=0

冋

P关␴ = 共0,i,T兲兴␶关␴ = 共0,i,T兲兴

冉冊

1 ␮ ⬇ ␭ ␭M共S − i兲 Finally, we obtain

冉冊

1 S log ␭M S−T

冉兺

T−1 k=0

when

M Ⰷ T,

共50兲

365

0 = − 关k1共s1 − i兲 + k2s2兴共m − i兲qi,0 + k1共s1 − i + 1兲共m − i + 1兲qi−1,0 ,

3P

0 = − 关k1s1 + k2共s2 − j兲兴共m − j兲q0,j

␮ ␭

366 367

− 1 = − 共k1ms1 + k2ms2兲q0,0 ,

Interestingly, in Eq. 共51兲, the term in vanishes, and thus we recover the zero order approximation Eq. 共19兲 for the MTT when particles cannot escape. In Fig. 3, the analytical formula 共49兲 and the result of Brownian simulations show reasonable agreement. It might be tempting to believe that replacing degradation by a small absorbing window would give similar results. It does not. Indeed, the probability to reach a window in a sphere containing several others depends nonlinearly on their distribution, through their capacitance 关17–19兴. Thus, the rates ␭k,m and ␮k,m will differ from the ones we obtained here. The geometrical configuration of

+ k2共s2 − j + 1兲共m − j + 1兲q0,j−1 ,

368 369 370 371 372

RE

pT1,j共⬁兲 = k1共s1 − T1 + 1兲共m − j − T1 + 1兲qT1−1,j ,

373

pi,T2共⬁兲 = k2共s2 − T2 + 1兲共m − i − T2 + 1兲qi,T2−1 ,

374

0 = 关k1共s1 − i兲 + k2共s2 − j兲兴共m − i − j兲qi,j + 共m − i − j + 1兲

375

⫻关k1共s1 − i + 1兲qi−1,j + k2共s2 − j + 1兲qi,j−1兴 . ,

376

If ␶共T1兲共respectively, ␶共T2兲兲 is the first time the threshold T1 377 1-7


p˙i,j = 关k1共s1 − i兲 + k2共s2 − j兲兴共m − i − j兲pi,j + 共m − i − j + 1兲 and the time-integrated equations are

␭ Ⰷ ␮.

362

364

⫻关k1共s1 − i + 1兲pi−1,j + k2共s2 − j + 1兲pi,j−1兴.

共51兲

329 330 331 332 333 334 335 336 337 338 339 340

冊

1 1 . + S−k S−i

361

p˙i,T2 = k2共s2 − T2 + 1兲共m − i − T2 + 1兲pi,T2−1 ,

00

␶T共␭, ␮,T兲 ⬇

册

1 ␮ 1− , ␭M共S − k兲 ␭共S − k兲

360

363

70

328

k=0

␮ ␭共S − k兲

359

p˙T1,j = k1共s1 − T1 + 1兲共m − j − T1 + 1兲pT1−1,j ,

]1

327

⬇ 1− 兺

T−1

71

326

冋

T−1

+ k2共s2 − j + 1兲共m − j + 1兲p0,j−1 ,

06

325

P关␴ = 共0,T兲兴␶关␴ = 共0,T兲兴

共53兲 348

we propose to estimate the probability that the threshold T1 of MS1 bindings is reached before that the number of bound MS2 reaches the threshold T2. We will also compute the corresponding conditional MTT. When T1 + T2 ⱕ m 共for T1 + T2 ⱖ m, the thresholds could possibly not be reached兲, the analysis uses a two-dimensional Markov chain. Adapting Eq. 共24兲, the probability density function pi,j共t兲 for i bindings MS1 and j bindings MS2 共0 ⱕ i ⱕ T1 and 0 ⱕ j ⱕ T2兲, satisfies the Markov equation

[E

324

共52兲 347

M + S2→ MS2 ,

F

=

兺␴ ␶共␴兲P共␴兲

343

When m molecules M can bind to sites S1 共of number s1兲 344 with a rate k1 or to sites S2 共of number s2兲 with a rate k2 共no 345 346 backward rate兲,

.

311 312

␴

the holes will now influence the escape and binding rates, 341 and this effect should be studied carefully. 342


K. DAO DUC AND D. HOLCMAN 1

n−1

P␴ = 兿 f共ik,ik+1兲,

T2 = 5

0.9

共55兲 384

0.8

0.6

T =3 2

0.5

0.4

0.3

T2 = 2

0.2

0

T2 = 1

F

0.1

1

1.5

2

2.5

3

3.5

4

4.5

5

CO

T

1

(a) 6

PY

5

4

3

L1

2

06

1

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

k

k⬍n−1

and

1 k2共s2 − k兲 if 1+ k1共s1 − j兲

k = n − 1,

冧

共56兲 400

T2−1

兺 k=0

冋

兺

␴=共0=i0ⱕ. . .ⱕik+1=T 兲

册

P␴ .

1

00 1

k 2s 2 ⫻ k 1s 1 + k 2s 2

冊

冉

T1 + T2 − 1

冊冉

k

k 1s 1 k 1s 1 + k 2s 2

T1+T2−1−k

.

RE

冉

兺 k=T

3P

P共␶共T1兲 ⬍ ␶共T2兲兲 ⬇

共57兲 404

冊

405 406 407 408

k

共58兲

409

410

k

We remark that P depends on the ratio k12 only, but not on the 411 number m of M molecules provided that T1 + T2 ⱕ M. The 412 conditional MTT to reach the threshold T1 before T2 is 413

共54兲

␶共k1,k2,T1,T2兲 =

Using similar combinatorial considerations as in Sec. III A, the probability P␴ of a path ␴ = 共i0 , i1 , . . . , in兲 for 0 = i0 ⱕ i1 ⱕ . . . ⱕ in = T1 and 1 ⱕ n ⱕ T2 is

where 1-8


兿 j=i

399

1 k2共s2 − k兲 1+ k1共s1 − j兲

70

1

ik+1−1

]1

381 382 383

pT ,j共⬁兲. 兺 j=0

k

ik ⫽ ik+1

if

T1+T2−1

T2−1

380

兿 j=i

In Fig. 4 we plotted using formula 共57兲 the probability P关␶共T1兲 ⬍ ␶共T2兲兴 as a function of T1 for fixed T2. Using the argumentation of formula 共43兲, when T1 Ⰶ s1 and T2 Ⰶ s2, this probability can be approximated by

is reached 共respectively, T2, the probability that T1 is reached before T2 is P共␶共T1兲 ⬍ ␶共T2兲兲 =

ik+1−1

1 k1共s1 − ik+1兲 1+ k2共s2 − k兲

P关␶共T1兲 ⬍ ␶共T2兲兴 =

FIG. 4. 共Color online兲 Probability and Mean time to reach the threshold T1 before T2 in a microdomain where ligands bind with two kind of sites S1 and S2. Up: we plot the probability P关␶共T1兲 ⬍ ␶共T2兲兴 to reach the threshold as a function of T1 for different values of T2. We present the exact formula 共57兲共solid line兲 and results obtained by Brownian simulations 共dash line兲 in a circular domain of radius R with s1 targets S1 of size ⑀1 and s2 targets S2 of size ⑀2. Down: we plot the conditional MTT as function of T1 for T2 = 5. The parameters are S1 = S2 = 5, M = 10, ⑀1 = ⑀2 = 0.05 ␮m, R 1 = 1 ␮m, and D = 1 ␮m2 s−1. The rates k1 and k2 are given by ␶ R2 4␲R 1 where ␶ = D 关log共 ⑀ 兲 + 4 兴共see 关15兴兲. We ran 500 simulations. 378 379

if ik = ik+1

385 386 387 388 389 390 391 392 393 394 395 396 397 398

where the third case corresponds to ik+1 − ikMS1 bindings fol- 401 lowed by a MS2 single event. Summing over all possible 402 403 paths,

T1

(b)

冦

1 k1共s1 − ik兲 1+ k2共s2 − k兲

71

0 0.5

f共ik,ik+1兲 =

[E

Conditional mean time to reach T1

where i1 counts the last bindings between M and S1, before any previous bindings between M and S2. In general, ik is defined such that ik+1 − ik is the number of MS1 bounds following exactly k-th MS2 bounds. If i1 = i2, there is another consecutive bound between a single M and another single S2 molecules, while for i1 ⬍ i2, there are i2 − i1 bindings between M and S1, followed by a single binding between M and S2 and so on. For k ⬍ n − 1, f共ik , ik+1兲 are the transition probabilities between the states 共兩MS1兩 , 兩MS2兩兲 = 共ik , k兲 , 共ik+1 , k + 1兲. For the special case k = n − 1, f共in−1 , T1兲 is the transition probability between the states 共in−1 , n − 1兲 , 共in , n − 1兲. The computation of f goes as follow: for ik = ik+1, the transition probability is the one of a M molecule to bind to one of the s2 − ikS2 free binding sites where there are s1 − k free S1 sites, that is k2共s2−ik兲 k1共s1−k兲+k2共s2−ik兲 .

T =4 2

0.7

OO PR

Probability to reach T1

k=0

兺␴␶␴ P共␴兲 , P关␶共T1兲 ⬍ ␶共T2兲兴

共59兲

414 415


THRESHOLD ACTIVATION FOR STOCHASTIC CHEMICAL… n−1

␶␴ = 兺 g共ik,ik+1兲,

416 417

共60兲

k=0

and where

if

OO PR

冦

共1/k2兲共m − ik − k兲共s2 − k兲

g共ik,ik+1兲 =

ik+1

共1/k1兲兺 j=ik 共m − j − k兲共s1 − j − k兲

+

共1/k2兲共m − ik+1 − k兲共s2 − k兲

ik+1−1

兺 j=i

ik ⫽ ik+1

共1/k1兲 if 共m − j − k兲共s1 − j − k兲

冧

FIG. 5. 共Color online兲 The Markov diagram for a path going from 共k0 , m + 1兲 to 共k2 , m兲.

k⬍n−1

and

CO

k

if

F

418

ik = ik+1

k = n − 1. k

Following the computation of Sec III B, for k12 , m Ⰷ 1, we ␶共T1 , T2兲, as 关see Eq. 共50兲兴 420 approximate the conditional MTT ¯ 419

P关␴ = 共0,T1兲兴␶关␴ = 共0,T1兲兴 T1−1

兺 i=0

⬇ 422

1 k2 s2 − k1m共s1 − i兲 k1 k1m

⫻

i=0

1 s1 − i

2

P关␴ = 共0, j,T1兲兴␶关␴ = 共0, j,T1兲兴

425

s2 k2 ⫻ k1 m共s1 − j兲

T1−1

427

⬇

兺 i=0

册

共61兲

vanishes and finally

1 k1m共s1 − 1兲

1 s1 log k 1m s1 − T1

冊

when

k 1 Ⰷ k 2,

m Ⰷ 1.

462 463 464 465 466 467 468 469 470 471 472

429 430 431 432 433

This result can be compared to formula 共51兲, obtained for a single threshold and a uniform killing rate. In both cases, we recover the zero order approximation of Eq. 共19兲, which relies asymptotically on MS1 properties 共and not the killing effect or the number of 兩MS2兩 bindings兲. V. CONCLUSION AND PERSPECTIVES

435 436 437 438 439 440 441 442

We have presented here a general approach to estimate the mean time to form exactly T chemical bonds between substrate and ligand particles. The present computation is based on a Markovian model, which allows us to obtain explicit formula, confirmed by Brownian simulations. Interestingly, the MTT is a nonlinear function T. This result may have other several consequences in cellular biology, including the design of molecular switches 关20兴 where upon the activation

qk1,m␭共S − k1兲共m − k1兲

␭共S − i兲兿关␭共S − k2兲 + ␮兴共m − k2兲 i=k1+1 ␭共S − i兲 + ␮ + 共terms not depending on qk1,m兲,

474

共A1兲 475

where the terms that do not depend on qk1,m will arise from the summation of k , m + r where r ⱖ 1 that will be taken off later on. The contribution of qk1,m+1 to qk1,m 共middle box in Fig. 5兲 is given by qk1,m =

473

k2−1

RE

434

qk2,m =

3P

共62兲

00

428

476 477 478 479

␭共S − k1 + 1兲 ␮ qk ,m+1 + qk −1,m . ␭共S − k1兲 + ␮ 1 ␭共S − k1兲 + ␮ 1 共A2兲 480

Similarly to Eq. 共A1兲, we get the contribution of qk0,m+1 to 481 qk1,m+1 共red box in Fig. 5兲. Finally, the expression for the 482 1-9


We prove relation 共34兲. Indeed, the formula is true for k = 1 or m = M − 1 and a direct computation shows it satisfies 共31兲. We obtain the formula, by considering Eq. 共31兲: qk,m can be expressed as a sum of two contributions, one depending on qk−1,m and the other on qk,m+1. Thus for k ⬍ k⬘ and m⬘ ⬍ m, we can express qk⬘,m⬘ as a function of qk,m, by summing the contribution of q共k⬘ , m⬘兲 in the diagram of Fig. 2 for all “paths” leading from 共k , m兲 to 共k⬘ , m⬘兲. In Fig. 5, we represent a possible path from 共k0 , m + 1兲 to 共k2 , m兲, when the killing happens at state 共k1 , m + 1兲. The contribution of qk1,m to qk2,m is obtained by using inductively 共31兲共left box in Fig. 5兲. Thus

70

冉

兺 i=0

1 . s1 − i

461

]1

¯␶共T1,T2兲 ⬇

k2 k1

T1−1

APPENDIX A

71

The first order term in

冋

1 + s1 − j

,

06

⬇ 426

冉兺冊册 T1−1

443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460

L1

424

i=0

1 + 共s1 − i兲2

[E

冋兺

T1−1

423

PY

421

of a molecular threshold, the cell fate can be reprogrammed. Another example is the nonlinear relation between the number Calmodulin Kinase II molecules and the induction of long term potentiation 关4兴 in synaptic plasticity. A final example is the spindle checkpoint occurring during the cell mitosis 关21兴: indeed during metaphase, centrosome nucleated microtubules interact with the chromosome kinetochores to build the mitotic spindle. Only after all chromosomes have become aligned at the metaphase plate, when every kinetochore is properly attached to a bundle of microtubules, does the cell enter anaphase. Our analysis can be used to estimate the probability and the mean time of the number of bindings occurring between kinetochores 共the number of which is equal to that of chromosomes ranging from a few to less than 50兲 and the anaphase activators 共our M-molecule兲. Finally, the present approach disregards the target organization in the cellular domain, which may have a significant effect, especially when the targets can cluster 关17–19兴.



冉

M−T

XT 兺

k=0

冊

T−1+k k

M

共1 − X兲k = 兺

k=T

冉冊

M k X 共1 − X兲 M−k . k 共A6兲 505

OO PR

Equivalently, we have to show 共replace X by 1 − X, divide by 506 507 XT and replace k by k − T on right side兲

兺 k=0

兺冉

M−1−T

CO

兺 k=0

冉冊

M X M−k−T共1 − X兲k . k+T

k

冊

M−1−T

Xk =

兺 k=0

冉冊 M−1 k+T

X M−1−k−T共1 − X兲k , 共A8兲 512

␮␭共S − k0兲共m − k0兲关␭共S − k1兲 + ␮兴关␭共S − k2兲 + ␮兴共m − k2兲 ⫻

册

␭共S − i兲 qk0,m+1 ␭共S − i兲 + ␮

+ sum of other contributions.

486

and let us prove 共A8兲 for M: First, we rewrite the left-hand 513 side of Eq. 共A7兲 514 M−T

兺 k=0

[E

冋兿

i=k0+1

册

␭共S − k0兲共m − k0兲 ␮ 兺 ␭共S − i兲 + ␮ 关␭共S − k2兲 + ␮兴共m − k2兲 0 i=k0

⫻

490

i=k0+1

册

␭共S − i兲 qk0,m+1 . ␭共S − i兲 + ␮

册

兺 k=0

冉冊兺冉冊冊兺冉

Xk +

M−1

M−T

X M−T .

共A4兲

k=1

=

k=0

Using that q0,M = 1 / 共M共␭S + ␮兲兲关Eq. 共26兲兴, we obtain the formula 共34兲.

502

APPENDIX B

503 504

We prove here relation 共42兲 is equal to Eq. 共43兲: ∀M ⱖTⱖ1

518

冉冊

M M M−T X M−k−1−T共1 − X兲k+1 + X . k+T+1 T

冉冊冊册

=共1 − X兲M−T +

+

500 501

冉冊

共A10兲 519

Using Pascal’s rule,

册

共A5兲

517

M M M−T X M−k−T共1 − X兲k + X k+T T

M−T−1

499

516

M X M−k−T共1 − X兲k k+T M−T

␭共S − i兲 M共␭S + ␮兲 q0,M . ␭共S − i兲 + ␮ 关␭共S − k兲 + ␮兴共m − k兲

冉

M−1 k+T+1

M−T

M−T−2

−X

k=0

M−T−2

+

M T

k=0

520 M−T−2

X M−T + 共1 − X兲 ⫻

兺 k=0

冋冉冊 M−1 k+T

X M−k−1−T共1 − X兲k ,

冉冊冊兺冉冊兺冉

=共1 − X兲


k

冊冉冊

M M−T + X + T

M−1 k+T

M−1

k+T+1

M−T−2

兺 k=0

冉冊 M−1 k+T

X M−k−1−T共1 − X兲k

共A11兲

RE

i=0

j

T−1+k

3P

⫻

M 0−m

冉

00

冋兿 1

k−1

␮

兿 ⱕk i ␭共S − i j兲 + ␮

兺 k=0

70

兺 0ⱕi ,..,i

M−1−T

Xk =

]1

冋

M−T

=

More generally, by summing over all the contributions, we can obtain an expression for qk,m as a function of q0,M : each contribution in the diagram corresponds to a certain path going down M − m times and moving k times forward 共see Fig. 6兲. The path goes down on the diagram at i1 , ..i M−m where 0 ⱕ i1 ⱕ . . . ⱕ i M−m ⱕ k. When summing over all configurations 共i j兲, we obtain qk,m =

冊

Second, we consider the right-hand side

71

冋兿

k2−1

k

06

冋

T−1+k

共A9兲 515

L1

qk2,m =

k2

冉

共A3兲

487 We obtain all the contributions by summing over all possible 488 k1 in the diagram, i.e., all k0 ⱕ k1 ⱕ k2 and we obtain

498

T−1+k

k=0

PY

qk2,m =

485

491 492 493 494 495 496 497

X =

contribution of the path represented in Fig. 5 is

k2−1

489

k

冊

M−T k

We prove 共A7兲 by induction: for T = 1 共A7兲 is true for all M 509 and for all M it is true for T = M − 1. 510 Suppose for all T that it is true for M − 1 ⱖ T, i.e., 511

F

484

T−1+k

共A7兲 508

FIG. 6. Markov diagram for a path starting at 共0 , M兲 and ending at 共k , m兲. There are k + M − m steps going down M − m times and going forward k times. 483

冉

M−T

521

522

X M−k−1−T共1 − X兲k 523

524

X M−k−1−T共1 − X兲k+1 ,

共A12兲 525


THRESHOLD ACTIVATION FOR STOCHASTIC CHEMICAL…

冊兺冉冊兺冉冊兺冉冉冊冊兺冉冊兺冉冊兺冉冉冊 M−T−1

M−1

= 526

k+T

k=0

M−T−2

−X

527

528

M−1

k+T+1

k=0

M−T−1

530

M−1 k+T

k=0

M−T−2 k=0

M−T−3 k=0

M−1

k+T+1

+ 共1 − X兲 M−T +

X

T

M−T−1 k=0

兺 k=0

冉冊冉冊 M−1 k+T

X M−k−T共1 − X兲k +

X M−k−T共1

X M−k−1−T共1 − X兲k +

冉冊

M M−T X T

k+T

M M−T X − 共1 − X兲 M−T−1 , T

536

共A16兲 537

X M−T ,

M−1

534

M M−T X , 共A15兲 T 535

X M−k−1−T共1 − X兲k +

冉冊 M−1 T−1

X M−T . 共A17兲 538

k

By induction, and with Eq. 共A9兲 it follows

X M−k−1−T共1 − X兲k+1 + X共1 − X兲M−T−1

M−T

兺 k=0

冉

T−1+k k

冊

M−T k

X =

兺 k=0

539

冉冊

M X M−k−T共1 − X兲k . k+T

共A14兲

共A18兲 540

L1

[E

541 542 543

University Press, New York, 2007兲. 关13兴 D. Holcman and Z. Schuss, J. Stat. Phys. 117, 191 共2004兲. 关14兴 A. Singer, Z. Schuss, D. Holcman, and B. Eisenberg, J. Stat. Phys. 122, 437 共2006兲. 关15兴 A. Singer, Z. Schuss, and D. Holcman, J. Stat. Phys. 122, 465 共2006兲. 关16兴 A. Singer, Z. Schuss, and D. Holcman, Narrow Escape III 122 491 共2006兲. 关17兴 S. Pillay, M. J. Ward, A. Peirce, and T. Kolokolnikov, SIAM Multiscale Modeling and Simulation 共to be published兲. 关18兴 A. Cheviakov, M. J. Ward, and R. Straube, SIAM Multiscale Modeling and Simulation 共to be published兲. 关19兴 A. Cheviakov, M. J. Ward, Optimizing the Principal Eigenvalue of the Laplacian in a Sphere with Interior Traps, special issue of Mathematical Methods and Modeling of Biological Sciences, held in Rio de Janiero 共March 2009兲, p. 21. 关20兴 M. Tigges and M. Fussenegger, Curr. Opin. Biotechnol. 20, 449 共2009兲. 关21兴 A. Ciliberto and J. V. Shah, EMBO J. 28, 2162 共2009兲.

RE

3P

00

70

]1

71

关1兴 C. R. Doering and J. C. Gadoua, Phys. Rev Lett. 69, 2318 共1992兲. 关2兴 M. Bier and R. D. Astumian, Phys. Rev Lett. 71, 1649 共1993兲. 关3兴 D. S. Bredt and R. A. Nicoll, Neuron 40, 361 共2003兲. 关4兴 J. Lisman, Philos. Trans. R. Soc. Lond. B Biol. Sci. 358, 829 共2003兲. 关5兴 L. Wolpert, Trends Genet. 12, 359 共1996兲. 关6兴 S. K. Reddy, M. Rape, W. A. Margansky, and M. W. Kirschner, Nature 共London兲 446, 921 共2007兲. 关7兴 F. Stegmeier, M. Rape, V. M. Draviam, G. Nalepa, M. E. Sowa, X. L. Ang, E. R. McDonald III, M. Z. Li, G. J. Hannon, P. K. Sorger, M. W. Kirschner, J. W. Harper, and S. J. Elledge, Nature 446, 876 共2007兲. 关8兴 R. Zwanzig and A. Szabo, Biophys. J. 60, 671 共1991兲. 关9兴 H. Zhou and A. Szabo, J. Phys. Chem. 100, 2597 共1996兲. 关10兴 D. Holcman and Z. Schuss, J. Chem. Phys. 122, 114710 共2005兲. 关11兴 N. G. Van Kampen, Stochastic Processes in Physics and Chemistry, 3rd ed. 共North-Holland, 2007兲. 关12兴 S. Redner, A Guide to First-Passage Processes 共Cambridge

06

544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 AQ: 559 #1 560 561 562 563

共1 − X兲

k+T

M−1

−

=

M−1

兺 k=0

=

共A13兲

PY

+

k+T

M−k−1−T

冉冊冉冊冊兺冉 M−T−1

CO

−X 531

k+T

k=1


M−1

M−1

− X兲k +

F

=

k+T

X M−k−1−T共1 − X兲k+1 + 共1 − X兲 M−T

M−T−2

X M−k−1−T共1 − X兲k −

M−T−2

M M−T + X − 共1 − X兲 M−T−1 , T

529

M−1

k=0

OO PR

+

533

=


k+T

k=0

M−T−2

532

M−1

冊兺冉冊兺冉

M−T−1



564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582

PROOF COPY [EL10671] 170003PRE NOT FOR PRINT!

FOR REVIEW BY AUTHOR

NOT FOR PRINT!

AUTHOR QUERIES —

#1

AU: Please verify Ref. 10 - we were unable to link with Vol. 1 - page 122. We found volume 122 page 114710. Is this the article?

F

OO PR PY

CO RE

3P

00

70

]1

71

06

L1

[E 1-12 PROOF COPY [EL10671] 170003PRE