如何用方程式量化metabolic.pdf

如何用方程式量化Metabolic pathway ?—part 2Biochemical systems theory (BST)• 生物化學系統理論(biochemical systems theory: BST)可 以對生物中的生合成代謝途徑進行系統模擬及分析• BST理論已經發展40多年，其用動態微分方程組描述系統 中變數對時間的變化情形• BST系統流入或流出通用方程式：1,i jn mf iii jVrX+==∏考慮生合成途徑中的中間物流出和流入方向，BST方程組可 以修改為不同形式，其中包括generalized mass action (GMA) system 及S-systemiiiXVV+−=−&For i =1,2,…..n,X1X2X31,i jn mf iii jVrX+==∏GMA model V.S. S system• GMA model differs from the S-system model only at the branch point.11ijijn mn mgh iijij jjXXXαβ++===−∏∏&S systemGMA system12 12 111......ijijijkn mn mn mfff iijijikj jjjXXXXγγγ+++====++++∏∏∏&iiiXVV+−=−&For i =1,2,…..n,12,...1(,,,....)iinnn mVVXXXXX−− ++=12,...1(,,,....)iinnn mVVXXXXX++ ++=S systemsX1X2X3Constant influx iiiXVV+−=−&For i =1,2,…..n,X1X2X3Constant influx 13112122111311323 22132221322(,)(,)()ggghgXVXXXXXVXXVXXXXααβαβ−+−=−=−=−=−&&,12,121211212112(,,...,,...):......(,,...,,...):......i n miiini n miiingggg innn minn mhhhh innn minn mVXXXXXXXXXVXXXXXXXXXαβ+++ +++− +++iiiXVV+−=−&S-system, where S refers to synergism and saturation of the investigated system.Note: all S system equations have the same mathematical form but differ in their parameter. 11ijijn mn mgh iijij jjXXXαβ++===−∏∏&For i =1,2,…..n,S systemsGMA model V.S. S system• GMA model differs from the S-system model only at the branch point.11ijijn mn mgh iijij jjXXXαβ++===−∏∏&S systemGMA system12 12 111......ijijijkn mn mn mfff iijijikj jjjXXXXγγγ+++====++++∏∏∏&iiiXVV+−=−&For i =1,2,…..n,X1X2X3Constant influx 13112122111311323 22132221322(,)(,)()ggghgXVXXXXXVXXVXXXXααβαβ−+−=−=−=−=−&&Pathway with branch points: X4 is a constant influx, please write the S-system equation for this pathway: X2X3X4X11313121411122122223133331111234112312341123222212222122233331333313334(,,)(,,),(,)(),(,)(),constant.ghgghhgghgghXVVVXXXVXXXXXXXXXXVVVXXVXXXXXVVVXXVXXXXXαβαβαβ+−+−+−+−+−+−=−=−=−=−=−=−=−=−=−=&&&S-systemPathway with branch points: X4 is a constant influx, please write the S-system equation for this pathway: X2X3X4X113113133121411212221222231333311112341212131322221222333313334,,,constant.gffggffgghgghXVVXXXXXXXXVVXXXXVVXXXXαγγαβαβ+−+−+−=−=−−=−=−=−=−=&&&GMA-systemX2X3X4X11313121411122122223133331111234112312341123222212222122233331333313334(,,)(,,),(,)(),(,)(),constant.ghgghhgghgghXVVVXXXVXXXXXXXXXXVVVXXVXXXXXVVVXXVXXXXXαβαβαβ+−+−+−+−+−+−=−=−=−=−=−=−=−=−=−=&&&S-SystemGMA-System13111313312114111212221222231112131123412121313112341212131322221222212223333133331(,,)(,)(,),(,)(),(,)()gfffgffgghgXVVVVXXXVXXVXXXXXXXXXXVVVXXVXXXXXVVVXXVXXγγγαβα+−−+−−+−+−+−+−=−−=−−=−−=−=−=−=−=−=&&&13333 3334,constant.ghXXXβ−=PLAS Software• PLAS = Power Law Analysis and Simulation available at: http://www.dqb.fc.ul.pt/docentes/aferreira/plas.html• Depensent variable: Xi,會隨系統動態改變 • Independent variable:同一實驗下為常數 （不同實驗可能有不同常數值）• 典型的生合成途徑中，中間產物會是隨時 間變化的依賴變數，酵素和受質濃度則為 獨立變數。

024012X1X2X3Branched Pathways•File: BranchS.plc •=================•Branched Pathway as S-system model•X1' = 0.8 X2^g12 X3^g13 X4^.5 - 4.959813621 X1^0.615672498 X2^-0.053731001 X3^- .092537998 •X2' = 3 X1^0.5 X2^-.1 - 1.5 X2^0.5•X3' = 2 X1^0.75 X3^-.2 - 5 X3^0.5•X1 = 0.5 •X2 = 0.5 •X3 = 1 •X4 = 0.25 •X5 = 0.5•g12=-1 •g13=-1•V1 = 4.959813621 X1^0.615672498 X2^- 0.053731001 X3^-.092537998 •V2 = 3 X1^0.5 X2^-.1 •V3 = 2 X1^0.75 X3^-.2Branched Pathways•File: BranchG.plc •=================•Branched Pathway as GMA model•X1' = 0.8 X2^g12 X3^g13 X4^.5 - 3 X1^0.5 X2^-.1 -2 X1^0.75 X3^-.2 •X2' = 3 X1^0.5 X2^-.1 - 1.5 X2^0.5 •X3' = 2 X1^0.75 X3^-.2 - 5 X3^0.5•X1 = 0.5 •X2 = 0.5 •X3 = 1 •X4 = 0.25 •X5 = 0.5•g12=-1 •g13=-1•Fluxes, defined as “transformations“ •V1 = 3 X1^0.5 X2^-.1 + 2 X1^0.75 X3^-.2 •V2 = 3 X1^0.5 X2^-.1 •V3 = 2 X1^0.75 X3^-.2024012X1X2X3S-System vs. GMA024012X1X2X3024012X1X2X3Mechanistic aspects may affect the parameter values, but do not influence the structure of the equation.如何求解? MATLAB BST toolboxDynamics of metabolite pools in L. lactis, derived from 20 mM [6- 13C]glucose metabolised under aerobic conditions at pH 6.5X2X3X4X11313121411122122223133331111234112312341123222212222122233331333313334(,,)(,,),(,)(),(,)(),constant.ghgghhgghgghXVVVXXXVXXXXXXXXXXVVVXXVXXXXXVVVXXVXXXXXαβαβαβ+−+−+−+−+−+−=−=−=−=−=−=−=−=−=−=&&&S-SystemGMA-System13111313312114111212221222231112131123412121313112341212131322221222212223333133331(,,)(,)(,),(,)(),(,)()gfffgffgghgXVVVVXXXVXXVXXXXXXXXXXVVVXXVXXXXXVVVXXVXXγγγαβα+−−+−−+−+−+−+−=−−=−−=−−=−=−=−=−=−=&&&13333 3334,constant.ghXXXβ−=何種系統比較好求解？Reversible PathwaysX1X2X3v12v23v21v32X4v41v141114111223212222141141221111141122122123322221222334)(),)(),constant.constant.gghhhgghXvvvvVVXXXXXvvvvVVXXXXXXαβαβ+−+−=−−−=−=−=−−−=−=−==&&&（（Condensation of poolsX2X3X4X5X1X1X5...X5X1What happens to -steady state? -dynamics?X2X3Linear pathway: X4 is a constant influx, please write the S- system equation for this pathway:X4X11411212232331111411141122221222122333323332334()(),()(),()(),constant.ghghghXVVVXVXXXXVVVXVXXXXVVVXVXXXXαβαβαβ+−+−+−+−。