Cellular Metabolic Models

Transcription

1 Cellular Metabolic Models. Cellular metabolism. Modeling cellular metabolism. Flux balance model of yeast glycolysis 4. Kinetic model of yeast glycolysis

2 Cellular Metabolic Models Cellular Metabolism

3 Basic Concepts Intracellular & extracellular phases extracellular» Intracellular inside a cell» Extracellular outside the cells intracellular Cellular growth» Growing cells consume nutrients from and secrete products to the extracellular enironment» Diision: one cell two cells» cells + substrates more cells + products

4 Cellular Metabolism The sum of all chemical changes that take place in a cell through which energy and basic components are proided for essential processes, including the synthesis of new molecules and the breakdown and remoal of others (National Cancer Institute).

5 Cellular Metabolism is Complex

6 Metabolic Pathways Cellular metabolism is accomplished through an interconnected series of metabolic pathways Characteristics» Initial substrates are products of other pathways or are externally supplied and transported across the cell membrane» These substrates are conerted to multiple metabolites through series of reactions» Final products are used as substrates for other pathways, accumulate inside the cell, or are secreted across the cell membrane

7 Organization of Metabolic Pathways

8 Biochemical Reactions S E P, S S E E M P, S E + M S E E E P P + P Substrate (S): consumed by the reaction Product (P): produced by the reaction Metabolite (M): substrate or product of reaction Enzyme (E): protein that catalyzes the reaction Reaction may inole multiple substrates, enzymes, or products; can be irreersible or reersible

9 Glycolytic Pathway Conerts glucose to pyruate Generates ATP & NADH ATP dries energy consuming reactions NADH dries reduction reactions Glucose + NAD + + ADP + Pi pyruate + NADH + H + + ATP + H O

10 Cellular Metabolic Models Modeling Cellular Metabolism

11 Leel of Modeling Detail Biomass Lumped description (unstructured) Detailed description (structured) Increasing metabolic detail

12 Structured Metabolic Models General characteristics» Mechanistic description of cell growth and product formation rates» Detailed modeling of intracellular reactions Adantages» Sound theoretical basis» Superior predictie capabilities» Extensible to new enironmental conditions Model types» Flux balance models require reaction stoichiometry» Kinetic models also require enzyme kinetics

13 S d[ M ] dt Kinetic Models = a µ [ M ] Stoichiometric coefficients (a,a )» Determines number of product molecules produced per molecule of substrate consumed» Often known from biochemistry literature Reaction rates (, )» Requires knowledge of enzyme kinetics (i.e. how reaction rates depend on metabolite concentrations)» Also called reaction flux Growth rate (µ)» Cellular growth causes dilution of metabolite concentrations M a P

14 The General Case M i N- N Intracellular mass balances d[ M dt i ] N d[ M] = aij j ([ M], K,[ M n]) µ [ M i ] = A([ M]) µ [ M] j= dt» [M]: n-dimensional ector of metabolite concentrations» : N-dimensional ector of fluxes; accounts for both reaction fluxes & membrane transport fluxes» A: nxn matrix of stoichiometric coefficients Sole ordinary differential equation system for unknown metabolite concentrations [M]

15 Flux Balance Models Stoichiometric equations assuming intracellular steady state and neglecting dilution: A =» : n-dimensional ector of intracellular fluxes» m balanced metabolites» A: mxn matrix of stoichiometric coefficients Flux balance model formulation» Gien a set of measured/specified fluxes ( m ) A = Acc + Amm = Acc = Amm b» b: m-dimensional ector of measured transport rates & fluxes» Sole stoichiometric model for unknown fluxes ( c )» If the A c is square and nonsingular: c = A c b

16 Comparison of Metabolic Models Flux balance models Assume an intracellular steady state Only require stoichiometric data Unknowns are the reaction fluxes The number of unknown fluxes should equal the number of balanced metabolites Easy to sole the linear algebra problem Kinetic models Capture dynamic intracellular behaior Require stoichiometric & enzyme kinetic data Unknowns are the metabolite concentrations No restrictions on the number of fluxes ersus the number of balanced metabolites Must sole coupled set of nonlinear differential equations

17 Metabolic Models Flux Balance Model of Yeast Glycolysis

18 Simplified Picture of Yeast Glycolysis extracellular intracellular glycerol NAD + NADH 6 glucose i glucose GP/DHP ATP ADP NAD + NADH acetaldehyde/ pyruate NADH NAD + acetaldehyde/ pyruate 4 o,-bpg ATP ADP 5 ethanol Wolf & Heinrich, Biochemical Journal, 45, -4,.

19 Steady-State Mass Balances Membrane transport ( i ) Glucose (extracellular) Glucose Reaction ( ) Glucose + ATP GP/DHP + ADP Reaction ( ) GP/DHP+NAD +,-BPG + NADH Reaction ( ),-BPG+ADP Acetaldehyde/Pyruate + ATP Reaction 4 ( 4 ) Acetaldehyde/Pyruate+NADH Ethanol + NAD + Reaction 5 ( 5 ) ATP ADP Reaction 6 ( 6 ) GP/DHP+NADH glycerol + NAD + Membrane transport ( o ) Acetaldehyde/Pyruate Acetaldehyde/Pyruate (extracellular) Glucose : = NADH : = i BPG : = 4 6 GP/DHP : = Acetaldeyde/Pyruate: ATP : = =

20 Linear System Representation Assume measurements of glucose influx i and acetaldehyde/pyruate efflux are aailable = o i

21 MATLAB Exercise Consider the flux balance model A = b: Use MATLAB to sole for the unknown fluxes gien:» i = 6, o =» i = 4, o = 4» i =, o = 6 = o i

22 Form Matrix and Check Rank >> A = [ ; - -; - ; - ; - -; - - ] A = >> rank(a) ans = 6

23 Compute Solutions >> b = [6 ]'; >> = linsole(a,b) = >> b = [4 4 ]'; >> = linsole(a,b) =

24 Compute Solutions cont. >> b = [ 6 ]'; >> = linsole(a,b) = First two solutions seem reasonable since all the computed fluxes are positie Implies that glucose is use to make pyruate That is glycolysis! Third solution seems unreasonable since some of the computed fluxes are negatie Implies that ethanol is used to make pyruate That is not glycolysis!

25 Metabolic Models Kinetic Model of Yeast Glycolysis

26 Simplified Picture of Yeast Glycolysis extracellular intracellular glycerol NAD + NADH 6 glucose i glucose GP/DHP ATP ADP NAD + NADH acetaldehyde/ pyruate NADH NAD + acetaldehyde/ pyruate 4 o,-bpg ATP ADP 5 ethanol Wolf & Heinrich, Biochemical Journal, 45, -4,.

27 Dynamic Mass Balances Membrane transport ( i ) Glucose (extracellular) Glucose Reaction ( ) Glucose + ATP GP/DHP + ADP Reaction ( ) GP/DHP+NAD +,-BPG + NADH Reaction ( ),-BPG+ADP Acetaldehyde/Pyruate + ATP Reaction 4 ( 4 ) Acetaldehyde/Pyruate+NADH Ethanol + NAD + Reaction 5 ( 5 ) ATP ADP Reaction 6 ( 6 ) GP/DHP+NADH glycerol + NAD + Membrane transport ( o ) Acetaldehyde/Pyruate Acetaldehyde/Pyruate (extracellular) ds Glucose : = i dt ds BPG : = dt dn NADH : = 4 dt 6 GP/DHP : Acetaldeyde/Pyruate: ATP : da dt ds dt = = ds dt 4 =

28 Enzyme Kinetics Redundant to model ADP and NAD + : - 6 : mass action kinetics: : mass action kinetics plus ATP inhibition: ) ( ) ( N S k A k N S k A A S k A S k N N S k N S k tot tot = = = = = = = + = q K I A A S k tot N tot A = + = + + NAD NADH ADP ATP

29 MATLAB Exercise Sole the kinetic model for the following two alues of the extracellular flux of acetaldehyde/pyruate:» o =» o = For each o alue, use the function fsole to find the steady-state solution and the function ode45 to integrate the model with the steady-state solution as the initial condition

30 Parameter Values Variable Symbol Value rate constant k mm - min - rate constant k 6 mm - min - rate constant k 6 mm - min - 4 rate constant k 4 mm - min - 5 rate constant k 5.8 min - 6 rate constant k 6 mm - min - Glucose uptake flux i mm/min Pyruate/acetaldehyde secretion flux o mm/min Total ATP+ADP A t 4 mm Total NADH+NAD + N t mm rate parameter K i.5 mm rate parameter q 4

31 MATLAB m-file function f = yeast_glycolysis(x) % Define model parameters q = 4; Ki =.5; At = 4; Nt = ; i = ; o = ; k = ; k = 6; k = 6; k4 = ; k5 =.8; k6 = ; % Define state ariables S = x(); S = x(); S = x(); S4 = x(4); N = x(5); A = x(6); % Calculate reaction rates = k*s*a/(+(a/ki)^q); = k*s*(nt-n); = k*s*(at-a); 4 = k4*s4*n; 5 = k5*a; 6 = k6*s*n; % Calculate deriaties f() = i-; f() = *--6; f() = -; f(4) = -4-o; f(5) = -4-6; f(6) = -*+*-5; f = f';

32 Solution for o = Find steady state for guess of for each ariable >> xss = fsole('yeast_glycolysis',[ ]) Equation soled. fsole completed because the ector of function alues is near zero as measured by the default alue of the function tolerance, and the problem appears regular as measured by the gradient. <stopping criteria details> xss = Define function handle to redefine number of function arguments to include time >> df yeast_glycolysis(x);

33 Solution for o = cont. Integrate model from steady state oer time interal of minutes >> [t,x]=ode45(df,[ ],xss); Plot time solutions for all ariables and just ATP >> subplot(,,) >> plot(t,x) >> ylabel('variable (mm)') >> xlabel('time (min)') >> legend('s','s','s','s4','n','a') >> subplot(,,) >> plot(t,x(:,6)) >> ylabel('atp (mm)') >> xlabel('time (min)') >> axis([,,,4]) Steady state is stable!

34 Solution for o = cont. 5 S Variable (mm) 5 S S S4 N A Time (min) 4.5 ATP (mm) Time (min)

35 Solution for o = >> xss = fsole('yeast_glycolysis',[ ]) xss = >> [t,x]=ode45(df,[ ],xss); >> subplot(,,) >> plot(t,x) >> ylabel('variable (mm)') >> xlabel('time (min)') >> legend('s','s','s','s4','n','a') >> subplot(,,) >> plot(t,x(:,6)) >> ylabel('atp (mm)') >> xlabel('time (min)') >> axis([,,,4]) Solution is oscillatory steady state is unstable!

36 Solution for o = cont. Variable (mm) S S S S4 N A Time (min) ATP (mm) Time (min)