Systematic model reduction techniques for chemically...

Systematic model reduction techniques for chemically reactive

systems Eliodoro Chiavazzo, Pietro Asinari

Multi-scale Modeling Lab, Department of Energetics, Politecnico di Torino

PHD COURSE: COMPLEX SYSTEMS IN ENGINEERING SCIENCES, Organizer: Prof. Nicola Bellomo, Deparment of Mathematics, Politecnico di Torino 21°-22° November 2011.

Brief outline • Detailed chemical kinetics and governing equations of reactive

flows;

• Combustion in homogeneous reactors;

• Model reduction and low dimensional manifolds;

• Invariant manifolds and construction techniques;

• A few words on model reconstruction for biological systems

• More references

Motivations • Energy conversion systems are today dominated and will for several

decades depend very significantly on combustion processes (still around 80%).

• Particularly in transportation systems (internal combustion engines and gas turbines), efficient and “near-zero” pollutant combustion processes can only be designed if complex reaction kinetics and their interaction with thermofluidics can be investigated and understood in depth.

The simplest fuel: H2

OHOH 2222

9 species, 21 reactions

+ Large disparity of time scales

Elementary reactions

OHHHO 2

• Detailed mechanisms aim at describing a chemical reaction at a more

fundamental level as it results from a large number of reversible collision

processes (elementary reaction steps) involving a usually remarkably large

number of chemical components (species):

Detailed chemistry modeling • The detailed description of combustion chemistry of practical fuels

(typically blends of higher hydrocarbons) can include hundreds of species participating in thousands of reactions.

)22(2 nnHC

Reactive flows: Governing equations

Conservation of mass,

NkYVVuxt

k ,...,1,,

pTVvhY

Conservation of species

Conservation of energy

Conservation of momentum

RTp Eq. of state

N. Fields at each grid node: Nf=3+D+N

Bell et al., PNAS 2005

N. of equations to solve at each time step:

Ex. dof107.1100256 93

A few examples: Chaotic laminar flames

Yu et al., MCS, Cagliari (Italy) 2011

Evolution of corrugated flame front:

Altantzis et al., Proc. Comb. Inst., 2011

Cellular structure of flames:

A few examples: Oscillating laminar flames

Pizza et al., Comb. Flame., 2008

Kerkemeier, PhD thesis ETH Zurich, 2011

A few examples: Turbulent flames Auto-ignition in turbulent non-premixed flows (hydrogen injected in co-flowing air, run on circa 130.000 CPUs):

Time scales in combustion

Starting point: Homogeneous reactors

k ,...,1,1

WTHcdt

Autonomous Dynamical system are to be solved. Two main drawbacks: 1) The number of degrees of

freedom is typically huge!

2) The dynamics is characterized by a wide range of time scales. This significantly hinders the numerical solution of them (due to stiffness)

• Those problems have been pointed out from the very beginning, and several methods aiming at simplifying the solution of the above governing equations have been investigated;

• Among others, the very popular techniques are surely: 1) the Quasi-Steady-State-Assumption – QSSA – (for fast species) and 2) the Partial Equilibrium Approximation (for fast reactions);

• Although those methods present a simple implementation, they are often out of reach for non-experts and not automated!!! (Chemist’s intuition needed).

Thermodynamic Lyapunov function

H,P,ϕ: fixed

Physical system: Homogeneous reactive mixture

Autonomous Dynamical system: Kinetic equations for a batch reactor

k ,...,1,1

WTHcdt

The Lyapunov function candidate L with respect to the above kinetic system of equations is suggested by the second law of thermodynamics. L can be constructed on the basis of the specific mixture-averaged entropy (mass based):

lnln1~

PRXRTSX

Elemental constraints • In a given batch reactor, the equilibrium point is unique. Notice that, the

kinetic system of equations typically has several zeros, however only one is

physically acceptable (positive concentrations):

• Due to conservation of the number of moles of “d” different atoms involved

in the reaction, elemental linear constraints have to be fulfilled:

• Hence, another Lyapunov function candidate G can be written as follows:

0Y,,,YYY eq

1 fYYfdt

k,...,1,

LddLYW

dClLL PHk

Phase-Space

• In homogeneous reactive mixtures, the state evolves in N-dimensional

phase space (N is the number of chemical species), however it is only a

sub-set of

• The reason for that are the elemental constraints (equality constraints) and

the positivity of the concentrations (inequality constraints) due to the fact

that negative concentrations are physically not acceptable:

,...,1,0

Phase-Space: Convex N-dimensional Polytope (hyper-polyhedron)

Steady-State

Simplification of a description 1/2

Steady-State

Initial conditions

k ,...,1,1

WTHcdt

Detailed description of the chemical phenomenon:

Simplified description of the same phenomenon (valid after a “long time”):

eq PHdL

HH,P,ϕ: fixed

Simplification of a description 2/2

k ,...,1,1

WTHcdt

Original problem: ODE’s system

Under which condition can we trust the above simplification (thermodynamic description)? t

• Thermodynamic data is reliable after a very long time!!!

• Finding the equilibrium state can be regarded as a simple model

reduction technique!!!

• However, the validity condition may be a rather restricting one!!!

eq PHdL

Reduced problem: Non-linear algebraic system

Dr Eliodoro Chiavazzo 18

Can we generalize that idea? • A useful generalization of the latter model reduction technique is

accomplished assuming the existence of hierarchical structures in the

phase-space due to disparate time scales in the dynamics, as exemplified

in the following cartoon: Arbitrary

initial conditions

Two-dimensional surface (2D manifold)

One-dimensional curve (1D manifold)

Steady State (0D manifold)

Solution trajectory (say 3D)

0tLOW DIMENSIONAL MANIFOLDS:

Low dimensional manifolds • Constructing low dimensional manifolds (of various dimensions) allows to

generalize the idea of model reduction, whereas the higher the manifold

dimension the more time-scales are included in the reduced model.

• The entire solution trajectory is the detailed model (dashed line): full model

= maximal computational effort = best accuracy

0D manifold, valid for: t

...210 ttt

1D manifold, valid for: tt2

2D manifold, valid for: But necessary for:

tt121 ttt

Full dimension (3D), valid for: But necessary for:

10 ttt

Reduced degrees of freedom • In general, a state can evolve in the full phase-space (N-dimensions = N

degrees of freedom (DoF) to describe a given phenomenon),

• However, the occurrence of low dimensional manifolds (after a given time)

forces the states to evolve in low dimensional subspaces, which can be

therefore described by a fewer number “M” of degrees of freedom:

Hopefully M<<N.

• Fewer degrees of freedom to take into account means both a less

computational effort and a better understanding of a complex phenomenon,

without renouncing (after a given time) to a good accuracy.

Elapsed time 0 1t2t

Full model with N=3 DoF

Dynamics with M=2 DoF, 2D manidold

Dynamics with M=1 DoF, 1D manidold

M=0 DoF, Steady state

• Model reduction techniques typically operate a cut with respect to the

time scales to be represented.

General strategy • The problem of simplification of large ODE’s system (system of kinetic

equation) turns out to be the computation of low dimensional manifolds in

the phase-space:

N DoF N-1 DoF 1 DoF

“M=0” easy: Equilibrium point (Thermodynamic description)

Construction of a low dimensional manifold with a given dimension “M”.

Very fast time scales

Fast time scales

Intermediate Time scales

Fast/Slow motions have different directions in the phase-space:

Initial condition

Steady state

Separation of motions

How do we compute low dimensional manifolds for model reduction in

combustion?

Beyond Equilibrium (0D): 1-2-3…ND?

N DoF N-1 DoF 1 DoF

Fast time scales

Intermediate Time scales

0 Elapsed time

Quasi-Equilibrium manifold (QEM)

Based on the implication due to: Motion separation + Lyapunov function

tLtLtLtL

The conseguence is: The minimum of L constrained to the fast direction gives a point “around” the low dimensional manifold

0 1 2 3 3 2 1 0

Fast direction

Iso-lines of the Lyapunov function L (convex function)

The geometry of a QEM When searching for a QEM, we guess the fast directions and assume that have fixed inclination in the full phase-space:

The locus of the constrained minima is called QEM

Fast directions are assumed to be known and fixed in the entire phase-space

Constrained equilibrium point of L or QEM point

Does it remind you anything?

2211 YY

STATESTEADY

• The blue dashed line represents an

ideal fast direction;

• The blue line can be interpreted as

the locus where the following linear

combination has the constant value ξ:

• The quantity ξ is a “slow variable”

because it does not change during

fast movements;

• In general, more constraints can be

imposed.

• We need to guess one inclination of

the fast direction (1D manifold) and

fix the slow variable ξ :

How do we find a QE-point?

,...,1,

,...,1,..

minimum~

Mathematically speaking, a QE-point is the solution of the problem:

PRXRTSX

lnln1~

This ensures that the number of atoms are conserved!!!

This imposes that the slow variables ξj are conserved during the fast motions

The locus of all the QE-points is called QE-manifold (considering ξj as variable parameters). The number “M” of additional constraints dictates the dimension of the QE-manifold

How do we solve the above problem?

Global minimization of the following Lagrange function:

RCCE parameterization • A M-dimensional QE-manifold can be computed as soon as the following M

parameterization vectors are specified:

• In combustion problems, it is rather popular the RCCE (Rate Controlled

Constrained Equilibrium) parameterization (by Keck & Gillespie, 1971),

where the above vectors are linked to known slow variables.

• For example, the total number of moles is a slow variable due to the fact

that recombination/dissociation reactions (that make the total number of

moles to change) are quite slow:

Spectral Quasi Equilibrium • RCCE parameterization requires that slow quantities are know in advance

(there are some recipes).

• A more systematic parameterization is given by the Spectral Quasi

Equilibrium SQE (by Chiavazzo, Gorban, Karlin, 2007):

0Y,,,YYY eq

1 fYYfdt

System of kinetic equations:

Jacobian matrix J of the above system:

• According to the SQE, the parameterization vectors are given by the left

eigenvectors of the Jacobian matrix evaluated at the equilibrium point:

Slow Invariant Manifolds – SIM and construction techniques (for homogeneous reactors)

• Manifolds suitable for model reduction collect all the trajectories after a short

relaxation (fast motion);

• After reaching such manifolds, fast dynamics is exhausted and only the slow

dynamics is left;

• Hence, those manifolds can be termed as low-dimensional manifolds of the

slow motions;

• Trajectories, once attracted to the manifold, do not leave any more the manifold

• The latter properties is typically referred to as “Invariance Condition”:

Invariance condition

Fast relaxation toward the manifold

Slow dynamics toward the steady state (always on the manifold)

Slow Invariant Manifolds (Best for Model Reduction)

Invariant manifolds • A given manifold is termed invariant with respect to a dynamical system if,

starting from any of its points (as initial condition at time t=0), the solution

trajectory proceeds along the manifold without leaving it any longer at a future

time t>0:

0Y,,,YYY eq

1 fYYfdt

Initial Condition

Steady state

2D invariant manifold with respect to the dynamical system

How do we check if a given manifold is invariant?

Modern model reduction methods • Various modern model reduction methods are all about techniques aiming at

accurately compute slow invariant manifolds - SIM;

• Constructing SIM is everything but a simple task;

• Usually, only approximations are computed;

• Quasi-Equilibrium Manifolds are rough approximations of the SIM, since they

are typically not invariant:

Non invariant manifold:

Tangent space

Invariant manifold:

Using the invariance condition

MODEL REDUCTION OF N-dimensional systems

SEARCHING FOR SLOW-INVARIANT MANIFOLDS

HOW CAN WE FIND SIMs? • SIM are some very special INVARIANT MANIFOLDS;

• Hence, an option is to write the invariance condition in a form of an equation e

try to solve it: INVARIANCE EQUATION;

• However, this is not a trivial task because INVARIANCE IS ONLY

NECESSARY CONDITION FOR SIM!!!

• Notice, for example, that all trajectories are 1D invariant manifolds

• SIM are only a small subset of invariant manifolds.

• Model reduction is understood as the construction of SIM in the phase-space:

Invariant manifolds SIM

Davis-Skodje toy-model

• Simple system with two degrees of freedom (y1,y2). A reduced description with

one degree of freedom is required:

• The Davis-Skodje example is characterized by a Slow Invariant Manifold (SIM)

whose analycal form is explicitly known, thus can be used for testing SIM

construction methods.

Singularly perturbed systems

• The Davis-Skodje example is a classical example of singularly perturbed systems

(derivatives multiplied with small number) and can be rewritten as follows:

• At a glance, we expect that if ε is small, then the variable y2 will be

characterized by a fast dynamics compared to y1. In other words, the

dynamics of y2 is expected to be slaved by the dynamics of y1 on a larger time

scale.

Slaving and invariance condition

• The idea of slaving can be elucidated by saying that fast variables have not

independent dynamics in time. Mathematically speaking, this can be expressed

as follows:

tyyty 122

• Hence, time derivatives of fast variables can be computed applying the chain

rule (and recording the first equation of the D-S system):

• Upon substitution in the second equation of the D-S system:

INVARIANCE EQUATION

Invariance condition 1/2

• The invariance equation (IE) states that: For any dependence of the slow

variable y1, function y2(y1) has to be such that the above equation is satisfied;

• Importantly, in the invariance equation there are no more time-derivatives;

• A certain function y2(y1) has to be found to describe the SIM;

• The latter function has to satisfy the IE at least up to some orders of accuracy;

• The IE itself does not provide the unknown function y2(y1), due to the fact that

there are simply too many solutions. Additional hypothesis are to be done

before trying to solve it: IE only necessary condition!!!

Invariance condition 2/2

12: yy does not satisfy the invariance condition

21, yyy

12: yydoes satisfy the invariance condition

Chapman-Enskog solution to the IC

,...,2,1,0...,... )(

22 nyyyyy nn

• Chapman-Enskog expansion is a perturbation method that attempts a solution

of the invariance condition, by exploiting the smallness of ε and assuming the

following shape of the unknown solution function y2(y1):

• Where:

tyyy nn

......y

yyyyyyy

• Upon substitution into the invariance condition:

• IC:

Dynamics at different scales • Grouping together terms of the same order in ε:

dyy 0)1(

ny n 0)(

• The full summation can be computed and it gives the exact SIM equation (very special case, typically we stop after a few terms):

Exact expression for the SIM 41

Trajectories solutions in the phase-space

• Solution trajectories (blue circles) in the phase-space (y1,y2) of the D-S

example with ε=1/50 – by Euler scheme with constant dt=5e-3:

• Asymptotic solution y2(y1) (black curve): 1

CE method for combustion

• In combustion, CE method is typically adopted for analysis purposes of fairly

simple systems, mainly due to the fact that small parameters are not explicitly

expressed in the RHS of dynamical systems;

• Usually, dynamical systems are regarded as a black box, hence there is a need

of automated procedure for constructing numerical approximations of the SIMs.

HH,P,ϕ: fixed

k ,...,1,1

WTHcdt

Chemical system Black box (from commercial codes)

f Automated

reduction

Desired technological chain for realistic systems:

Method of Invariant Manifold

A.N. Gorban I.V. Karlin

• This approach is about an alternative method for

solving the invariance equation. Introduced (for

chemical kinetics) in 2004 by Gorban and Karlin

How do we write the invariance equation in a general form?

f • Define a Projector operator (matrix)

onto the tangent space, such that:

fPfPPPP

• General form of the invariance equation:

fPfTangent space

Leicester University (UK)

ETH-Zurich

General form of the IE • In general, invariance can be imposed by introducing a projector operator onto

the tangent space and writing down the following condition:

cfcPcf

Example: It does not satisfy the invariance equation (local condition):

NYYc ,,1

The idea here is to solve the (non-linear) Invariance equation in order to find SIM

How do we solve non-linear equations?

Local correction

Initial guess – Typically non-invariant manifold (E.G. QE-Manifolds)

Slow invariant manifold – to be found

Intermediate iteration

Steady state

Iterative solution of the IE • In the same spirit of the Newton-Raphson method for solving non-linear

equations, The Method of Invariant Manifold (MIM) operates a linearization of

the Invariance Equation (non linear because of both the Projector and the

vector field) around an initial guess and try to solve it.

0~~ 00000 ccJcfcPccJcffPf

Incomplete linearization of the IE:

Plus a solvability condition:

0~ 0 ccP

Plus the conservation of atoms

k,...,1,0

The MIM algorithm 1/2

ccJcfcPccJcf nnnnn

• Linear system to solve with respect to cn+1

Iterations terminate when the

invariance defect is small:

cfcfcPcffc ~

The MIM algorithm 2/2

ccJcfcPccJcf nnnnn

• Algebraic system of equations to solve at each node of the manifold:

• It proves convenient to introduce the following vectors:

Basis of the subspace: DP ker~

• The last two equations of (*) are automatically fulfilled if we impose:

• The sistem (*) can be recast as follows by scalar product with bk, to be solved

with respect to the coefficients δi:

hwith Being the dimension of DP ker~

hkbJPbbJP kki

i i ,...,1,~

Construction of the projector: An example (1D)

Unity vector:

uuffP ˆˆ~

Euclidean projector:

ˆˆˆˆ

ˆˆˆˆ~ yx uuu ˆ,ˆˆ Tensorial notation (N=2):

How do we construct an Euclidean 2D projector?

2D tangent space (hyper-plane)

Gram-Schmidt ortho-normalization:

112121

uuuuuu

ˆ,ˆu

2211ˆˆˆˆ

~uufuuffP

Relaxation of the Film Equation (FE)

A.N. Gorban I.V. Karlin U. Maas

• The Film equation is dynamical equation

written for manifolds, aiming at refining an

initial non-invariant manifold. Used by Maas

& Collaborators for Combustion problems,

and further developed by Gorban and Karlin.

12: YY

Initial guess – Typically non-invariant manifold (E.G. QE-Manifolds)

Hint: Write by yourself an explicit first-order scheme for solving the FE (e.g. for the Davis-Skodje model)!

Leicester University (UK)

ETH-Zurich University of Karlsruhe (Germany)

Example: H2 + Air

Chiavazzo, PHD thesis, ETH-Zurich 2009

Reduced system to solve (not closed):

ik YfMPt

NqM Matrix, q<<N

SIM do provide closures for reduced systems.

k ,...,1,

Detailed system:

Spectral decomposition of J 1/2

U. Maas S.B. Pope

• The Intrinsic Low dimensional Manifold (ILDM) method is a very popular method in combustion (introduced in 1992 by Maas and Pope) for computing accurate approximations of SIM, though typically NOT INVARIANT!!!

Basic idea: The slow invariant manifolds (SIM) represent the locus of points “c” in the phase-space where we observe very strong contractions along some directions (fast directions) compared to other transversal directions (slow directions along the SIM):

Perturbation along the fast direction:

cccJcfccf

ccJcfccf

The fast direction is almost an eigen-direction of the Jacobian matrix “J” evaluated at “c”.

University of Karlsruhe (Germany)

Cornell University (NY - US)

Perturbation along the slow direction:

cccJcfccf

ccJcfccf

The slow direction (SIM) is “almost” an eigen-direction of the Jacobian matrix “J” evaluated at “c”. Due to the disparate time scales, there exists a separation between “fast” and “slow eigenvalues”:

• The above idea can be generalized by noticing that it can be established a hierarchy of eigen-vectors on the basis of the corresponding eigen-values of J:

NMNMN vvvv

,,,,, 11

011 NMNMN

Fast subspace Slow subspace

Spectral decomposition of J 2/2

• The ILDM approach divides the eigenvectors of the Jacobian matrix “J” in two

sub-groups: The first one identifies the slow sub-space, whereas the other

identifies the fast sub-space.

• By decomposing the vector field “f” into components of the fast- and slow-

subspace, the equation of the ILD-Manifold is found imposing that the fast

component is null:

slowfast fff

21 vvfff slowfastslowfast

ILDM condition:

Non-Cartesian coordinate systems 1v

How do we find the components in a Non-Cartesian coordinate system?

ijji vv

Let’s introduce the dual spaces as follows:

2211 vfvvfvf

Kronecker delta

The ILDM equation

slowfast fff

21 vvf slowfast

00 1 fvfast

ILDM equation:

2211 vfvvfvf

ijji vv

Slow-Fast decomposition of the vector field “f”:

In general, let’s compute the matrix “Q” and its dual:

NMNMN vvvvQ

,,,,, 11

Fast Slow

0 fQ fast

ILDM equation:

Hint: Find the analytical ILDM of the Davis-Skodje model!

H. Lam

Computational Singular Perturbation (CSP) method

D. Goussis Princeton University (NJ - USA)

• Similarly to ILDM, the CSP method looks for a decomposition into fast and slow

modes of the right-hand side of the kinetic equation system “f”:

0Y,,,YYY eq

1 fYYfdt

MNf vvA

N x (N-M) Matrix of the fast vectors

NMNs vvA

1 N x M Matrix of the slow vectors

(N-M) x N Dual of Af: IAB f

M x N Dual of Af: IAB s

fBAfBAf s

The four matrices are updated according to the CSP iterative algorithm, and the SIM equation is:

Lebiedz’s Method

D. Lebiedz University of Freiburg (Germany)

• This method is called Minimal Entropy Production

Trajectories (MEPT);

• It looks for approximations of SIM, and it typically

does not deliver the exact SIM;

• It has been introduced in 2004-2006;

• It is based on a variational problem: Minimization

of the Entropy production under constraints

0Y,,,YYY eq

1 fYYfdt

,...,1,

,...,1,0

An approximate SIM is given by all solution trajectories that minimize an objective function (time integral of a function ϕ).

Two options have been adopted:

Entropy production

Trajectory Curvature

Atom conservation

Parameterization

Kevrekidis’s method

I. Kevrekidis Princeton University (NJ - USA)

Reverse integration method:

Initial condition

• According to this approach, we let a system relax from an arbitrary initial condition;

• After some time, the trajectory is expected to land on the SIM (non simple to check it);

• From this point on (yellow), we can integrate the system backward and extend the manifold;

• Simple and fascinating idea, though with a quite tricky implementation when the dimension of the manifold is high;

• Still under investigations.

Forward integration

Backward integration Steady state

ICE-PIC Method

S. B. Pope Cornell University (NY - USA)

Z. Ren

• This method is called: Invariant Constrained

Equilibrium edge PreImage Curve method (ICE-

• This method is based on the notion of Quasi-

Equilibrium Manifold (Constrained Equilibrium

Manifold) with the RCCE parameterization and an

algorithm to improve it;

• It has been introduced in 2006 by Prof. Pope and

collaborators

QE-Manifold

Steady-State

Convex Polytope (Phase-Space) boundary

Back to the edge of the domain

Forward integration

CE-point

ICE-PIC point

• First, the Quasi-Equilibrium problem is solved and the CE-point found;

• Second, an appropriate boundary point “cbound” is found by the Preimage Curve Method;

• Finally, forward integration allows to find a much more accurate state than the CE point. boundc

Relaxation Redistribution Method Yet another (and numerically stable) way of solving the Film equation of dynamics, and the problem of minimal reduced description (choice of the manifold dimension q):

Chiavazzo & Karlin, PRE 2011

Global construction in the Phase-Space: Local construction in the Phase-Space:

http://arxiv.org/PS_cache/arxiv/pdf/1011/1011.1618v2.pdf

More info at:

Minimal description: Example

Chiavazzo & Karlin, PRE 2011

Homogeneous reactor: Detailed vs reduced with variable dimension:

Closure of reduced equations (flows)

FSIM for homogeneous reactor

Transport term

Chemistry term

Primitive variable

Collection of SIM for homogeneous reactor

Full dynamics (transport + chemistry)

Reduced dynamics (projection onto tangent)

Full dynamics

Reduced description of reactive flows

NqiYm k

ki ,...,1, Reduced observables

NkYYYFt

Yikiik

k ,...,1,, 2

Species equations

Energy eq. + Momentum Eq. + Eq. of state

kkk ,...,1,

Reduced species

equations

Energy eq. + Momentum Eq. + Eq. of state

REDUCTI ON

Typical strategy:

NOT CLOSED!

Closure provided by the collection of SIMs (typically stored in the form of a LUT)

Example: Detailed vs Reduced

Fiorina et al, Comb. Theo. Model. 2003

(A) Detailed solution;

(B) Reduced solution.

Methane: 29 species, 300 reactions

Example: Detailed vs Reduced

Gicquel et al, Comb. Theo. Model. 1999

13 species 2 species 13 species 2 species

Biological systems • Biological systems are often characterized by several interacting agents (biological

network) and processes with disparate time scales;

• Sometimes, model reduction techniques may be reversed in order to construct models for describing time evolution of e.g. biological systems (Model reconstruction);

• The simplest instance of bio-network (Micaelis-Menten catalytic reaction):

Network:

Mechanism:

Kinetic model:

An example

Schematics: Sugar from CO2 Modeling: Agents interacting in a network

An approach to dissipative bio-systems

Basic assumption for dissipative bio-systems:

We assume that dynamics is described by a hyerarchy of relaxation processes towards low dimensional manifolds, and each process can be approximated by BGK (Bhatnagar–Gross–Krook) – like expressions:

Fasano, Chiavazzo, Asinari, Unpublished 2011

Gibbs free energy (ideal systems):

Dynamical system: Approximation of SIMs (QEM approximation):

Parameter tuning

Chose parameters: Mi , That ensure minimal deviation from experimental data

Mitogen-activated protein kinase (MAPK)

Optimized system:

Fasano, Master thesis, Politecnico Torino 2011

http://www.ebi.ac.uk/biomodels-main/

MAPK: Network inference

MORE INFO AT…

http://e-collection.library.ethz.ch/view/eth:41898?q=chiavazzo

Useful links:

http://angkinetics.narod.ru/pdf/CiCP2007vol2_n5_p964.pdf

http://www.global-sci.com/freedownload/v8_701.pdf

Gorban&Karlin 2005, Springer

http://www.springerlink.com/content/x737842j7438mu30/

http://iopscience.iop.org/1478-3975/8/5/055011

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