The mean time to reach a threshold MTT is the mean first passage time for the number of bound ... several efforts were made 8,9 to model chemical reactions.
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
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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,
PROOF COPY [EL10671] 170003PRE PHYSICAL REVIEW E 81, 1 共2010兲
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 + ... kk−1k−2k−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 kk−1 k=2 kk−1k−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+ 22 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
PROOF COPY [EL10671] 170003PRE
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
PROOF COPY [EL10671] 170003PRE PHYSICAL REVIEW E 81, 1 共2010兲
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
PROOF COPY [EL10671] 170003PRE
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
PROOF COPY [EL10671] 170003PRE PHYSICAL REVIEW E 81, 1 共2010兲
THRESHOLD ACTIVATION FOR STOCHASTIC CHEMICAL… 1
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
PROOF COPY [EL10671] 170003PRE
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
PROOF COPY [EL10671] 170003PRE PHYSICAL REVIEW E 81, 1 共2010兲
K. DAO DUC AND D. HOLCMAN
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
PROOF COPY [EL10671] 170003PRE
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
PROOF COPY [EL10671] 170003PRE PHYSICAL REVIEW E 81, 1 共2010兲
THRESHOLD ACTIVATION FOR STOCHASTIC CHEMICAL… 309
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
PROOF COPY [EL10671] 170003PRE
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
PROOF COPY [EL10671] 170003PRE PHYSICAL REVIEW E 81, 1 共2010兲
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
PROOF COPY [EL10671] 170003PRE
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 4R 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
PROOF COPY [EL10671] 170003PRE PHYSICAL REVIEW E 81, 1 共2010兲
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
PROOF COPY [EL10671] 170003PRE
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.
PROOF COPY [EL10671] 170003PRE PHYSICAL REVIEW E 81, 1 共2010兲
K. DAO DUC AND D. HOLCMAN
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
1-10 PROOF COPY [EL10671] 170003PRE
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
PROOF COPY [EL10671] 170003PRE PHYSICAL REVIEW E 81, 1 共2010兲
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.
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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
X M−k−1−T共1 − X兲k
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
=
X M−k−1−T共1 − X兲k
k+T
k=0
M−T−2
532
M−1
冊 兺 冉 冊 兺 冉
M−T−1
X M−k−1−T共1 − X兲k
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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?
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