1、uni7CFBuni7EDF生物学数学uni57FA础前言uni4EC0么是uni7CFBuni7EDF生物学?“Systems biology is the science of discovering, modeling, understanding and ultimately engineer-ing at the molecular level the dynamic relationships between the biological molecules that define livingorganisms.”Leroy Hood, Ph. D., M.D.,Presiden
2、tInstitute for Systems BiologyAll biological phenomena, whether its digestion of a sugar molecule, beating of the human heart,or neutralizing an invading virus, are the result of complex systems. Thus our approach is to focusresearch on biological systems as a whole, rather than pursue the tradition
3、al approach of focusing onindividual genes, proteins, or parts of an organism.uni7CFBuni7EDF生物学uni7684研究uni5185uni5BB9?uni7CFBuni7EDF生物学研究uni65B9uni6CD5?Scientists from multiple disciplines (biology, chemistry, mathematics, physics, etc.) work closelytogether to fully understand all aspects of the i
4、nherently complex systems intrinsic to living organisms.Such in-depth understanding is ultimately essential to realizing our goal of predictive, preventive,personalized medicine.uni7CFBuni7EDF生物学与数学? “uni5B83(物理学)uni7684范畴uni53EFuni5B9A义为我uni4EEC全uni90E8uni77E5uni8BC6中uni80FD够用数学uni8BED言表uni53D1uni7
5、684uni90A3个uni90E8分”爱uni56E0uni65AF坦 uni7EDFuni8BA1，模uni578B，分uni6790，模uni62DF收uni96C6uni7ECFuni9A8C数据 a45总uni7ED3uni5B9A性、半uni5B9A性uni89C4uni5F8B a45uni5EFAuni7ACB数学模uni578Ba63求uni89E3、uni53D1展数学理uni8BBAa27用uni7ECFuni9A8C资uni6599uni9A8Cuni8BC1模uni578Ba64a64a64a64a73a0a0a0a0a18“Most readers of this
6、publication will know that post-genomics and proteomics are phrasesthat mean little that is specific but herald an encyclopaedic era of information about the way biologicalcells and their genes and proteins behave. But how best to make sense of it all? It is, at last, possibleto anticipate mathemati
7、cs becoming useful in the modelling of the systems.”Nature 407 2000, 819.uni5185uni5BB9uni7B80uni4ECB? uni57FAuni56E0表uni8FBE uni57FAuni56E0调uni63A7网uni7EDC Toggle switches 生物振荡2uni7CFBuni7EDF生物学数学uni57FA础 生命uni8282uni5F8B uni80DAuni80CEuni53D1uni80B2 uni7EC6uni80DE分uni88C2与分化 uni7CFBuni7EDF生物学前uni6
8、CBFuni4ECBuni7ECD uni968F机uni8FC7程(Master equation, Langevin equation, Fokker-Plank equation) uni5FAE分uni65B9程(uni5EFA模，uni5B9A性理uni8BBA，数值求uni89E3) uni968F机uni5FAE分uni65B9程(uni5EFA模，数值求uni89E3，稳uni5B9A性分uni6790) uni53CDuni5E94扩散uni65B9程(uni5EFA模，数值求uni89E3)补充阅uni8BFB材uni6599:1. Mackey, M. C., Santi
9、llan, M., Mathematics, Biology, and Physicss: Interactions and interepen-dence, Notices AMS, 52(2005)(8).2. Sontga, E. D., Molecular systems biology and dynamics: an introduction for non-biologists.3. Alon, U., An introduction to systems biology, Chapman i= 1, ,N). uni8FD9uni91CCvji 0 表示uni53CDuni5E
10、94Rjuni4EA7生分子Si, vji 0)表示uni53CDuni5E94Rj 在下个uni65F6uni95F4区uni95F4t,t+ uni5185uni53D1生uni7684次数. uni56E0为uni6BCF次uni8FD9样uni7684uni53CDuni5E94uni90FDuni628A分子Si uni7684个数uni589Euni52A0vji, uni7CFBuni7EDF中分子Si 在uni65F6刻t+ uni7684个数为Xi(t+) =xt,i +Msummationdisplayj=1Kj(xt,)vji, (i = 1, ,N). (1.4.17)
11、 eq:1.4:1uni8FD9uni91CC, Kj(xt,) 是uni968F机uni53D8uni91CF. uni8981uni5F97到uni5BF9所有 uni7684uni7CBE确uni63CFuni8FF0, 我uni4EEC需uni8981求uni89E3化学主uni65B9程. 然而, 我uni4EECuni53EFuni4EE5在下面uni7684条uni4EF6下uni7ED9uni51FAuni5F88好uni7684uni8FD1似.条uni4EF6一: uni9996先, uni53D6 充分小,使uni5F97在uni65F6uni95F4区uni95F4t,
12、t+ uni5185, uni7CFBuni7EDFuni7684uni72B6态uni53EA有uni5FAE小uni7684改uni53D8, uni56E0此, 所有uni7684propensity function uni51E0乎uni4FDD持不uni53D8:aj(X(t) aj(xt), t t,t+,j 1,M. (1.4.18) eq:1.4:2通常地, uni6BCF次uni53CDuni5E94uni90FDuni53EA使uni67D0uni79CD分子地个数uni589Euni52A0或uni51CF少1 , 所uni4EE5, 当uni7CFBuni7EDF地u
13、ni53CDuni5E94物地数uni91CFuni8FDC大uni4E8E1 uni65F6, uni53EAuni8981uni53D6 充分小, 上面uni7684条uni4EF6一是uni5F88uni5BB9易uni6EE1uni8DB3uni7684.根据条uni4EF6一,在uni65F6uni95F4区uni95F4t,t+uni5185uni53D1生uni7684所有uni53CDuni5E94uni90FD不改uni53D8uni7CFBuni7EDFuni7684propensity function. uni56E0此,所有uni53CDuni5E94在uni65F6
14、uni95F4区uni95F4t,t+uni53D1生uni7684uni6982uni7387uni53EFuni4EE5uni8BA4为是uni76F8uni4E92uni72ECuni7ACBuni7684.uni56E0此, Kj(xt,)等uni4E8E当propensity func-tion等uni4E8Eaj(xt) uni65F6, uni53CDuni5E94通道Rj 在uni65F6uni95F4 uni5185uni7684uni53D1生次数. uni8FD9个次数uni6EE1uni8DB3uni72ECuni7ACBPossion分布Pj(aj(xt),).5un
15、i7CFBuni7EDF生物学数学uni57FA础uni8FD9uni91CCP(a,t) 表示当uni67D0个uni4E8Buni4EF6在uni4EFB意uni65E0穷小uni4E8Buni4EF6区uni95F4dtuni5185uni51FAuni73B0uni7684uni6982uni7387为adt是, 在长uni5EA6为tuni7684uni65F6uni95F4区uni95F4uni5185uni51FAuni73B0uni7684次数. uni4EE4Q(n;a,t) 表示P(a,t) 等uni4E8En (整数) uni7684uni6982uni7387, 则由关
16、uni7CFBQ(0;a,t+dt) =Q(0;a,t)(1adt)uni53EFuni4EE5uni5F97到Q(0;a,t)t = at, Q(0;a,0) = 1.由此uni5BB9易uni5F97到Q(0;a,t) =eat. uni5BF9uni4EFB意n 1, 根据uni6982uni7387uni7684乘uni6CD5uni5B9Auni5F8B, 由关uni7CFBQ(n;a,t) =integraldisplay tt=0Q(n1;a,t)adt Q(0;a,tt).通uni8FC7数学归uni7EB3uni6CD5, uni53EFuni4EE5uni5F97到一般un
17、i7684公式Q(n;a,t) = eat(at)nn! , (n = 0,1,2,).uni73B0在, 我uni4EECuni53EFuni4EE5uni8BA1uni7B97uni968F机uni53D8uni91CFP(a,t) uni7684均值uni548Cuni65B9uni5DEEP(a,t) = varP(a,t) =at.当at 1 uni65F6, uni53EFuni4EE5uni8BC1明eat(at)nn! (2piat)1/2 expparenleftbigg(nat)22atparenrightbigg.uni56E0此, 当at 1, uni968F机uni5
18、3D8uni91CFP(a,t) uni53EFuni4EE5由具有uni76F8同uni7684均值uni548Cuni65B9uni5DEEuni7684正态分布来uni8FD1似:P(a,t) N(at,at), if at 1. (1.4.19) eq:1.A3uni56E0此由条uni4EF6一, uni65B9程(1.4.17) uni53EFuni4EE5uni8FD1似为xi(t+) =xt,i +Msummationdisplayj=1vjiPj(aj(xt),), (i = 1, ,N). (1.4.20) eq:1.4:3条uni4EF6uni4E8C:uni65F6un
19、i95F4区uni95F4 充分大,使uni5F97在uni65F6uni95F4区uni95F4t,t+ uni5185uni53D1生uni7684化学uni53CDuni5E94uni7684次数uni7684期望值大uni4E8E1, 即Pj(aj(xt),) =aj(xt) 1, j 1,M. (1.4.21) eq:1.4:4uni5F88显然, uni8FD9个条uni4EF6uni548C条uni4EF6一是uni77DBuni76FEuni7684, uni53EFuni80FD会uni51FAuni73B0uni8FD9样uni7684uni60C5uni51B5: 两个条
20、uni4EF6uni65E0uni6CD5同uni65F6uni6EE1uni8DB3. 在uni8FD9uni79CDuni60C5uni51B5下,我uni4EECuni7684模uni578B不uni80FDuni6EE1uni8DB3. 但是, 在uni5F88多uni60C5uni51B5下, uni8FD9uni8FD9两个条uni4EF6是uni53EFuni4EE5同uni65F6uni6EE1uni8DB3uni7684,uni4F8Buni5982, 当uni53D1生uni7CFBuni7EDF中uni7684uni53CDuni5E94uni6BCFuni79CD分子u
21、ni7684个数uni90FDuni8DB3够大uni65F6. uni8FD9uni65F6aj(xt)是大数, 即使 uni5F88小, 上面uni7684条uni4EF6也是uni53EFuni4EE5uni6EE1uni8DB3uni7684.当条uni4EF6uni4E8Cuni6EE1uni8DB3uni65F6, 我uni4EECuni53EFuni4EE5uni628APossion 分布Pj(aj(xt),) uni8FD1似为具有uni76F8同uni7684均值uni548Cuni65B9uni5DEEuni7684正则分布. uni56E0此, 我uni4EEC由下面u
22、ni7684关uni7CFBxi(t+) =xt,j +Msummationdisplayj=1vjiNj(aj(xt),), (i = 1, ,N). (1.4.22) eq:1.4:5uni8FD9uni91CCN(m,2) 表示均值为m, uni65B9uni5DEE为2 uni7684正则分布. uni6CE8意到在uni8FD9uni91CC, 我uni4EECuni628A整数uni7684Possion 分布uni53D8成为uni8FDEuni7EEDuni5B9E数uni7684正则分布. uni8FD9样, 分子数Xi 也uni76F8uni5E94uni7684uni53
23、D8成为是uni5B9E数uni7684. uni53E6外, M 个正则分布是uni76F8uni4E92uni72ECuni7ACBuni7684. uni8FD9是uni56E0为我uni4EEC假uni5B9A所有uni7684Possion 分布Pj uni90FD是uni76F8uni4E92uni72ECuni7ACBuni7684.利用正则分布uni7684uni7B80单关uni7CFBN(m,2) = m+N(0,1),我uni4EECuni53EFuni4EE5uni628A(1.4.22) 改uni5199为uni53E6外uni7684形式:xj(t+) = xt,j
24、 +Msummationdisplayj=1vjiaj(xt) +Msummationdisplayj=1vjiaj(xt)1/2Nj(0,1), (j = 1, ,N). (1.4.23) eq:1.4:66uni7CFBuni7EDF生物学数学uni57FA础uni8FD9uni91CCuni7684正则分布Nj(0,1)uni90FD是uni72ECuni7ACBuni7684.下面, 假uni8BBE 同uni65F6uni6EE1uni8DB3条uni4EF6一uni548C条uni4EF6uni4E8C,并uni8BB0 为dt. uni53E6外, 我我uni4EEC用白噪uni
25、58F0j(t) uni8BB0uni6EE1uni8DB3uni72ECuni7ACB正则分布Nj(0,1) uni7684uni968F机uni53D8uni91CF. uni8FD9uni91CC, 白噪uni58F0uni6EE1uni8DB3关uni7CFBj(t) = 0, i(t)j(t) =ij(tt), i,j 1,M,t.并uni8BB0Xj(t) = xt,j, 则uni65B9程(1.4.23) uni53EFuni4EE5表uni8FF0为xi(t+dt) = xi(t) +Msummationdisplayj=1vjiaj(xt)dt+Msummationdispl
26、ayj=1vjia1/2j (xt)j(t)(dt)1/2, (j = 1, ,N). (1.4.24) eq:1.4:7引uni8FDBuni7EF4uni7EB3uni8FC7程(Winer process) Wj,使uni5F97dWj = Wj(t+dt)Wj(t) = j(t)(dt)1/2uni53EFuni4EE5改uni5199上面uni7684uni65B9程为dxi(t) =Msummationdisplayj=1vjiaj(xt)dt+Msummationdisplayj=1vjia1/2j (xt)dWj, (j = 1, ,N). (1.4.25) eq:1.4:8u
27、ni8FD9uni91CCdxj(t) = xj(t+dt)xj(t). uni8FD9个就是化学朗之万uni65B9程(Chemical Langevin Equation). 1.5 uni8BA1uni7B97模uni62DF前面我uni4EECuni4ECBuni7ECDuni4E86uni63CFuni8FF0化学uni53CDuni5E94uni7684uni51E0uni79CD数学模uni578B, 分别uni6D89uni53CA到常uni5FAE分uni65B9程, uni5DEE分uni65B9程(化学主uni65B9程),偏uni5FAE分uni65B9程(Fokker
28、-Plank uni65B9程), uni968F机uni5FAE分uni65B9程. uni8FD9uni4E9Buni65B9程uni7684uni8BA1uni7B97模uni62DF分别uni6D89uni53CA不同uni7684数学uni9886uni57DF, uni53EFuni4EE5uni53C2考uni76F8uni5E94uni7684数学专业教材. uni8FD9uni91CCuni7B80单uni4ECBuni7ECDuni5982下. 1.5.1 常uni5FAE分uni65B9程uni7684数值模uni62DFuni5DEE分uni6CD5, 软uni4EF6
29、: xppaut. 1.5.2 求uni89E3化学主uni65B9程Gilliespie uni7B97uni6CD5:1. 初uni59CB化Xi，并uni4EE4初uni59CBuni65F6uni95F4t = 0.2. uni8BA1uni7B97a( = 1, ,M)，并uni4EE4a0 = summationtextM=1a.3. uni4EA7生0,1 上uni7684平均分布uni968F机数r1 uni548Cr2, 并uni4EE4 = (1/a0)ln(1/r1)，uni53D6 为uni6EE1uni8DB3条uni4EF6summationtext1=1 1sec
30、.(1.6.34) uni7ED9uni51FAuni4E86在一uni6BB5uni65F6uni95F4uni5185(uni4F8Buni5982, 大uni4E8E1 sec), uni81EA由uni7684uni57FAuni56E0位uni70B9占总数uni7684百分uni6BD4与uni6291制子X uni7684浓uni5EA6uni7684关uni7CFB. 当X = Kduni65F6, %50 uni7684位uni70B9是uni81EA由uni7684.假uni8BBE当位uni70B9是uni81EA由uni7684uni65F6, uni76F8uni5E
31、94uni57FAuni56E0uni7684转录uni7387为. 则mRNA uni7684uni4EA7生uni7387(成为promoter activity) uni548Cuni6291制子X uni7684关uni7CFB为promoter activity = 1 +X/Kd. (1.6.35) eq:mm4uni8FD9uni91CCKd uni79F0为repression coefficient.uni73B0在, 我uni4EEC考虑uni53E6一uni79CDuni60C5uni51B5, X uni8981uni548Cuni8BF1uni5BFC物SX uni7
32、ED3合uni79F0复合体SXX uni4EE5后才有活性. uni5982uni679Cuni6BCF个Xuni53EAuni80FDuni7ED3合一个SX, 则有关uni7CFBX +SXX = XT其中XT 表示X uni7684总数. 假uni5B9AX uni548CSX uni78B0uni649E并uni7ED3合成复合体uni7684速uni7387常数为jon, 复合物uni7684分uni89E3速uni7387常数为joff. 则复合物uni7684uni52A8uni529B学uni65B9程(质uni91CF作用uni5B9A理)为dSXXdt =konXSX j
33、offSXX. (1.6.36) eq:m219uni7CFBuni7EDF生物学数学uni57FA础并且假uni5B9Auni8BF1uni5BFC物SX uni7684数uni91CF充分大, 其数uni91CFuni7684uni53D8化uni53EFuni4EE5uni5FFD略. 在平衡态uni65F6, 有关uni7CFBKXSXX = XSXuni8FD9uni91CCKX 是复合物SXX uni7684uni89E3uni79BB常数. 则复合物SXX uni7684数uni91CFuni548Cuni8BF1uni5BFC物uni7684数uni91CFSX uni7684
34、关uni7CFBuni53EFuni4EE5由Michaelis-Menten uni65B9程(也uni79F0为米氏uni65B9程)表示uni51FA来:XSX = XTSXSX +KX. (1.6.37) eq:mm11uni73B0在uni5982uni679CX 上有n个uni7ED3合位uni70B9, uni53EFuni4EE5同uni65F6uni548Cn个SX uni7ED3合成复合体nSXX 并且uni88ABuni6FC0活. 则有关uni7CFBnSXX+X0 =XT. (1.6.38) eq:mm16uni8FD9uni91CCX0 表示uni81EA由uni7
35、684X, 并且中uni95F4态(与X uni7ED3合uni7684uni8BF1uni5BFC物uni7684个数少uni4E8En 个)uni90FDuni5FFD略. 复合物nSXX uni7684形成是通uni8FC7X uni548Cn个SX 分子uni7684uni78B0uni649E而形成. uni8BBEuni53CDuni5E94速uni7387常数为jon, 则collision rate =jonX0SnX. (1.6.39) eq:mm12uni4EE4uni79BBuni89E3常数为joff:dissociation rate = joffnSXX. (1.6
36、.40) eq:mm13uni53C2数joff 通常uni5BF9uni5E94与X uni548CSX 之uni95F4uni8FDEuni63A5uni7684化学键uni7684强uni5EA6. 复合物nSXX uni7684uni52A8uni529B学uni65B9程为dnSXXdt = jonX0SnX joffnSXX (1.6.41) eq:mm14uni8FD9uni91CC假uni8BBEuni7EC6uni80DEuni5185S uni7684数uni91CFuni5F88大, 其数uni76EEuni7684uni53D8化uni53EFuni4EE5uni5FF
37、D略. 在平衡态uni7684uni65F6候, 有关uni7CFBjoffnSXX = jonX0SnX. (1.6.42) eq:mm15由关uni7CFB(1.6.38), 我uni4EEC有(joff/jon)nSXX = (XT nSXX)SnX.由此uni53EFuni4EE5uni5F97到uni7ED3合uni7684X 所占uni7684uni6BD4uni4F8BnSXXXT =SnXKnX +SnX (1.6.43) eq:mm17其中KnX = joff/jon. uni8FD9个就是Hill equation, uni7CFB数n通常uni79F0为是Hill uni
38、7CFB数(Hill coefficient). 当n 1 uni65F6,通常uni79F0为是合作uni7684.uni6CA1有uni7ED3合uni7684uni6291制子X uni7684浓uni5EA6是X0XT =11 +(SX/KX)n. (1.6.44) eq:mm18uni73B0在考虑uni53E6外uni7684uni60C5uni51B5,假uni8BBE存在uni8BF1uni5BFC物S. uni6291制子uni53EFuni4EE5uni548Cuni8BF1uni5BFC物uni7ED3合为复合体XSX. uni8BF1uni5BFC物通uni8FC7un
39、i548Cuni6291制子uni7ED3合, 阻止uni6291制子uni6291制uni57FAuni56E0uni7684表uni8FBE, uni4ECE而uni8BF1uni5BFCuni57FAuni56E0uni7684表uni8FBE. 此uni65F6, uni6291制子X uni53EFuni4EE5有三uni79CDuni72B6态:uni81EA由uni7684, 与DNA位uni70B9uni7ED3合, 或者与uni8BF1uni5BFC物uni7ED3合:XT = X0 +XD+nSXX. (1.6.45) eq:mm5uni8FD9样, 我uni4EECuni
40、5F97到下面uni52A8uni529B学uni65B9程:dXDdt = konX0DkoffXD, (1.6.46)dnSXXdt = jonX0SnX joffnSXX. (1.6.47)uni8FD9uni91CC, 我uni4EEC假uni8BBE复合物nSXX 不uni80FD与D uni7ED3合, 并且uni7EC6uni80DEuni5185SX uni7684数uni91CFuni5F88大, 其数uni76EEuni7684uni53D8化uni53EFuni4EE5uni5FFD略.在平衡态uni65F6, uni53EFuni4EE5uni5F97到关uni7CFB
41、KXnSXX = X0SnX, KdXD =X0D (1.6.48) eq:mm810uni7CFBuni7EDF生物学数学uni57FA础uni8FD9uni91CCKX =joff/jon 为uni89E3uni79BB常数(for lac repressor, KX 1M 1000 inducer (IPTG) molecules/cell).由上面关uni7CFB(1.6.45) uni548C(1.6.48) uni53EFuni4EE5求uni89E3uni51FAX0XT =11 +D/Kd +SnX/KX.由uni81EA由DNA 所占uni6BD4uni4F8B与uni81E
42、A由uni6291制子uni7684浓uni5EA6uni7684关uni7CFB(1.6.34), uni53EFuni4EE5uni5F97到promoter activity (uni8BB0为f =f(SX)uni548Cuni8BF1uni5BFC物SX 之uni95F4uni7684关uni7CFBf = 1 +(XT/Kd)/(1 +fDT/(Kd) +SnX/KX). (1.6.49) eq:mm19当SnX/KX DT/Kd uni65F6, 上面关uni7CFBuni53EFuni4EE5uni8FD1似为f = 1+ (XT/Kd)/(1 +SnX/KX). (1.6.5
43、0) eq:mm20上面关uni7CFBuni7ED9uni51FAuni4E86uni57FAuni56E0uni7684活性与uni8BF1uni5BFC物uni7684浓uni5EA6之uni95F4uni7684关uni7CFB. 当SX = 0 uni65F6, 有f(SX = 0) /(1 +XT/Kd). uni8FD9个也uni79F0为是是basal promoter activity, 表示当uni6CA1有uni8BF1uni5BFC物uni65F6uni7684promoter 活性. 当SX = S1/2 (XT/Kd)1/nKXuni65F6, uni57FAuni
44、56E0uni7684活性恢复uni8FBE到最大值uni7684一半(f = /2).uni73B0在考虑uni6FC0活子uni7684uni60C5uni51B5: uni53EA有当X uni7ED3合到uni57FAuni56E0位uni70B9D 上uni65F6, uni76F8uni5E94uni7684mRNA 才会uni88AB转录. 根据前面uni7684uni8BA8uni8BBA, uni57FAuni56E0uni7684promoter activity uni548Cuni6FC0活子uni7684浓uni5EA6uni7684关uni7CFBuni53EFun
45、i4EE5通uni8FC7Michaelis-Menten uni65B9程表示uni51FA来:promoter activity = XKd +X. (1.6.51) eq:mm10uni8FD9uni91CCX 表示具有活性(uni53EFuni4EE5uni548CDNA位uni70B9uni7ED3合) uni7684uni6FC0活子uni7684浓uni5EA6.uni5982uni679C存在uni8BF1uni5BFC物SX uni53EFuni4EE5uni548Cuni6FC0活子uni7ED3合(假uni8BBEuni6FC0活子存在n 个作用位uni70B9), un
46、i6FC0活子uni53EA有当与n 个uni8BF1uni5BFC物uni7ED3合为复合体nSXX后, 才有活性(uni8FD9uni91CCuni5FFD略中uni95F4uni72B6态, 即与X uni7ED3合uni7684uni8BF1uni5BFC物uni7684个数少uni4E8En 个uni7684uni60C5uni51B5). 此uni65F6, 有关uni7CFBX +nSX X, X +DD通前面uni7684uni8BA8uni8BBA, 有活性uni7684uni6FC0活子uni7684浓uni5EA6为X = nSXX = XTSnXKnX +SnX.uni
47、56E0此, uni57FAuni56E0uni7684活性uni548Cuni8BF1uni5BFC物uni7684浓uni5EA6uni7684关uni7CFB为f(SX) = XKd +X. (1.6.52) eq:mm22当SX = S1/2 = (Kd/XT)1/nKXuni65F6, uni57FAuni56E0uni7684活性uni8FBE到其最大值uni7684一半.一般地, uni5982uni679C一个uni57FAuni56E0uni65E2有uni6291制子, uni53C8有uni6FC0活子, uni57FAuni56E0uni7684活性uni7CFB数为f
48、(X1, ,Xm) =summationtextii(Xi/Ki)ni1 +summationtexti(Xi/Ki)mi (1.6.53) eq:mm22uni8FD9uni91CCXi 表示uni6291制子或者是uni6FC0活子uni7684浓uni5EA6, Ki 表示uni76F8uni5E94uni7684uni6291制或uni6FC0活uni7CFB数.11uni7CFBuni7EDF生物学数学uni57FA础 1.7 补充阅uni8BFB材uni65991. van Kampen, N. G. 1992. Stochastic process in physics and chemistry. North-Holland, Amster-dam, 1992.2. Gillespie, D. T. 1977. Exact stochastic simulation of coupled chemical reactions. J. Phys. Chem.81:2340-2361.3. Gillespie, D. T. 2000. The chemical Langevin equation. J. Chem. Phys. 113:297306.12第uni4E8Cuni7AE0 u
