Cmputatinal mdeling techniques Lecture 4: Mdel checing fr ODE mdels In Petre Department f IT, Åb Aademi http://www.users.ab.fi/ipetre/cmpmd/
Cntent Stichimetric matrix Calculating the mass cnservatin relatins Calculating the steady state Elementary fluxes Sensitivity cefficients 5..4 Advanced cmputatinal mdeling
Define all these cncepts in a general framewr, n the level f reactin-based mdels The discussin abut the mathematical details f these cncepts is fr ODE-based mdels Similar definitins can als be given fr ther math framewrs Recall T each reactin we assciate a reactin rate it tells hw fast reactants are cnsumed, prducts are prduced The frm f the reactin rate depends n the inetic law chsen fr the reactin Mass-actin Michaelis-Menten Inhibitin f varius types Hill 5..4 Advanced cmputatinal mdeling 3
Stichimetry The stichimetric cefficients dente the quantitative prprtin in which substrate and prduct mlecules are invlved in a reactin. Examples: Fr an irreversible reactin S +S P, the stichimetric cefficients f S, S, and P are -, -, and, respectively. In general, the stichimetric cefficients are psitive fr prducts and negative fr reactants. Fr a reversible reactin S +S P, the stichimetric cefficients f S, S, and P are -, -, and, respectively. Fr a reactin S +S P+S the stichimetric cefficients are -,, and, respectively. 5..4 Advanced cmputatinal mdeling 4
Stichimetric matrix The stichimetric cefficients f a reactin netwr cnsisting f s species and r reactins are rganized in a s-called stichimetric matrix, dented N=(n ij ) sxr, where n ij dentes the stichimetric cefficient f species S i in reactin R j. Example: Reactin netwr Stichimetric matrix r : A B r : A+C D r 3 : D B+E r 4 : A+B D+B A B C D E 5..4 Advanced cmputatinal mdeling 6
Stichimetric matrix Fr a reactin netwr cnsisting f s substances and r reactins the dynamics, in particular the change f cncentratins in time, is described by the fllwing system f equatins: S dt i = n ii v j fr all i s, where v j is the rate f reactin j d r j=, This can be rewritten in the matrix ntatin: ds dt = NN, where S is a vectr f the cncentratins f all the substances in the reactin netwr, i.e. S=([S ],,[S s ]) T, and v= (v,,v r ) T is the vectr f the reactin rates. 5..4 Advanced cmputatinal mdeling 7
Stichimetric matrix Example A B ( ) A+B C ( +, - ) Stichimetric matrix: Reactin rates (mass actin inetics): 8 = N [C] [A][B], [A] + = = ν ν = + [C] [A][B] [A] d d[c] d d[b] d d[a] t t t 5..4 Advanced cmputatinal mdeling
Stichimetric matrix The stichimetric matrix cntains valuable infrmatin abut the structure f the netwr Mass cnservatin relatins Steady states Elementary fluxes Sensitivity cefficients Discuss each f them in the rest f this lecture 5..4 Advanced cmputatinal mdeling 9
Mass-cnservatin relatins 5..4 Advanced cmputatinal mdeling
Mass cnservatin relatins Frequently, the cncentratins f several species invlved in bichemical reactin netwrs are included in s-called cnservatin sums. A characteristic feature f such species is that their gain and lss rates are equal; they can frm cmplexes with ther species r be part f ther species. A mass cnservatin relatin is a cnstant linear cmbinatin f sme f the species f the mdel 5..4 Advanced cmputatinal mdeling
Mass cnservatin relatins Example reactins: A A A + B A :B A :B C + A :B C N = species: A, A, B, A :B, C - The ttal amunts f A and B are cnserved in time. Neither f them is prduced nr degraded. #A + #A + #A :B = cnst. #B + #A :B = cnst. 5..4 Advanced cmputatinal mdeling
Mass cnservatin relatins T identify the cnservatin relatins we slve the fllwing equatin in matrix G: where N is the stichimetric matrix Indeed, fr such G: Example (cntinued): 3 GN = =. = GNv dt ds G GN G N S = = = = C B : A B A A 5..4 Advanced cmputatinal mdeling
Recall frm linear algebra The number f independent rws f G, i.e. the number f cnservatin relatins, is equal t s-ran(n). In the example s=5 and Ran(N)=3. It fllws that G cntains independent rws, i.e., there are tw mass cnservatin relatins. Observatin: if the stichimetric matrix is full ran, it fllws that the system has n cnservatin relatins. 5..4 Advanced cmputatinal mdeling 4
Mass cnservatin relatins Cnservatin relatins can be used t reduce the system f differential equatins ds/dt=nv describing the dynamics f a reactin netwr. Each cnservatin relatin leads t ne mre dependent variable, that can be expressed in terms f the independent variables and eliminated frm the system f ODEs 5..4 Advanced cmputatinal mdeling 5
Mass cnservatin relatins Example reactins: A A A + B A :B A :B C + A :B C The mass cnservatin relatins (based n G): [A]+[A ]+[A :B]=K [B]+[A :B]=K In ther wrds: [A]=K-[A ]-[A :B] [B]=K -[A :B] The initial system f 5 ODEs in [A], [A ], [B], [A :B], [C] is reduced t a system f 3 independent ODEs in [A ], [A :B], [C] 6 5..4 GN G N S = = = = C B : A B A A Advanced cmputatinal mdeling
Steady states 5..4 Advanced cmputatinal mdeling 7
Steady state Steady state ne f the basic cncepts f dynamical systems thery, extensively utilized in mdelling. Steady states (statinary states, fixed pints, equilibrium pints) are determined by the fact that the values f all state variables remain cnstant in time: S(t)=S If S()=S, then ds dt = NN = Slve the equatin in the unnwn S (a vectr with s cmpnents, ne fr each variable) N is the stichimetric matrix v is a functin f the cmpnents f S An algebraic (system f) equatin(s) fr the typical inetics, e.g. mass-actin r MM 5..4 Advanced cmputatinal mdeling 8
Steady state Example (mass actin inetics) A B ( ) A+B C ( +, - ) Steady state algebraic equatins ([A], [B], and [C] are unnwns) r 9 = + ] [ ] [ ] [ ] [ C B A A = + = + = + + + [C] [B] [A] [C] [B] [A] [A] [C] [B] [A] [A] N ν 5..4 Advanced cmputatinal mdeling
Elementary fluxes 5..4 Advanced cmputatinal mdeling
Elementary flux mdes Cncept f elementary flux mde a minimal set f enzymes (r, in ther wrds, reactins) that can perate at steady state the smallest sub-netwrs that allw a binetwr t functin at steady state a minimal cmbinatin f reactins whse cmbined effect maintains the netwr in steady state any subset f it des nt maintain the steady state they ffer a ey insight int the bjectives f the netwr each elementary flux mde shuld have a clear bilgical interpretatin in terms f the bjectives f the netwr determines whether a given set f enzymes/reactins are feasible at steady states Larger flux mdes can be btained by cmpsing several flux mdes: steady-state flux distributins 5..4 Advanced cmputatinal mdeling
Calculating the elementary flux mdes We are interested in linear cmbinatins f reactins whse cmbined effect is t preserve the steady state dente w i the weight f reactin i in the flux mde Recall: ds dt f fluxes = NN, where N is the stichimetric matrix and v is the vectr We are interested in cmbinatins f fluxes (w,,w r ) that ensure dd dd = In ther wrds, slve the equatin Nw= in the unnwn w Recall frm linear algebra: the slutin is called the ernel (r the null space) f matrix N In general, the slutin is a vectrial space we are interested in its base; all ther slutins are linear cmbinatins f the elements in the base 5..4 Advanced cmputatinal mdeling
Example v v v 3 S S v 4 S 3 Stichimetric matrix: NK= yields slutin K = v 4 = means that in any steady state, the rates f prductin and degradatin f S 3 must be equal 5..4 Advanced cmputatinal mdeling 3
Sensitivity cefficients 5..4 Advanced cmputatinal mdeling 6
Lcal sensitivity analysis Lcal sensitivity analysis is a methd t estimate the changes brught int the system thrugh small perturbatins in the parameters f the mdel. It prvides means t: estimate the rbustness f the mdel against small changes in the mdel, identify pssibilities fr bringing a certain desired change in the system. We write the system f ODEs describing the dynamics f a reactin netwr in the fllwing frm: where κ=(,, M ) T is the parameter vectr. 5..4 Advanced cmputatinal mdeling 7
Lcal sensitivity analysis First rder lcal cncentratin sensitivity cefficients: hw the slutin depends n small variatins in the parameter values We dente S(t,κ)=([S ](t,κ),[s ](t,κ),,[s N ](t,κ)) T the slutin f the system with respect t the parameter vectr κ. The cncentratin sensitivity cefficients are the time functins fr all i N, j M. 5..4 Advanced cmputatinal mdeling 8
Lcal sensitivity analysis Very ften hwever, the fcus is n sensitivity analysis arund steady states. In case f asympttically stable steady states, cnsider lim t ( S/ j )(t) statinary sensitivity cefficients. reflects the dependency f the steady state n the parameters f the mdel. 5..4 Advanced cmputatinal mdeling 3
Scaled sensitivity cefficients When used fr cmparing the relative effect f a parameter change in tw r mre variables, the sensitivity cefficients must have the same physical dimensin r be dimensinless. One simply cnsiders the matrix C f (dimensinless) nrmalized (als called scaled) sensitivity cefficients: Numerical estimatins f the nrmalized sensitivity cefficients fr a steady state may be cmputed in sftware applicatins such as COPASI (http://www.cpasi.rg) r SBML-SAT (http://sysbi.mlgen.mpg.de/sbml-sat), a tl fr MATLAB (http://www.mathwrs.cm/). 5..4 Advanced cmputatinal mdeling 33
Nte: a similar analysis can als be dne fr the dependency with respect t the initial cnditins Sip it here 5..4 Advanced cmputatinal mdeling 34
Lcal sensitivity analysis example Mdel: A -> B, B -> A ODEs: d[a]/dt = - [A] + [B] d[b]/dt = [A] - [B] Numerical setup: [A]() = mml/ml, [B]() = mml/ml, =.5 s -, =.3 s - Steady state cncentratins: [A] ss = 7.5 mml/ml, [B] ss =.5 mml/ml 5..4 Advanced cmputatinal mdeling 36
Lcal sensitivity analysis Scaled statinary sensitivity cefficients: C ij [A] -.646.64766 [B].374766 -.374859 Interpretatin: increasing frm.5 by % yields a decrease in the level f [A] at the steady state by ~.6%. Nte: these results are accurate nly fr infinitesimally small changes! 5..4 Advanced cmputatinal mdeling 37
Lcal sensitivity analysis Verificatin (% change): set t.55 (increase by % with respect t the riginal value) new steady state: [A] new_ss =7.4534 mml/ml, [B] new_ss =.5466 mml/ml ([A] new_ss -[A] ss )/[A] ss =7.4534/7.5-».993789-» -.6 = -.6% ([B] new_ss -[B] ss )/[B] ss =.5466/.5-=,378-».37% T be cmpared with C = -.646(%) and C =.374766(%) Verificatin (% change): set t.33 (increase by % with respect t the riginal value) new steady state: [A] new_ss =7.958 mml/ml, [B] new_ss =.48 mml/ml ([A] new_ss -[A] ss )/[A] ss =7.958/7.5-»,64-=6,4% ([B] new_ss -[B] ss )/[B] ss =.48/.5-=,963856-=-3,6% T be cmpared with C = 6.4766(%) and C = -3.74859(%) 5..4 Advanced cmputatinal mdeling 38