analysis of biochemical reaction network systems using ...pogudin/workshop/samal.pdfsatya swarup...

Analysis of Biochemical Reaction NetworkSystems Using Tropical Geometry

Satya Swarup Samal

Joint Research Center for Computational Biomedicine (JRC-COMBINE)RWTH Aachen University

Workshop on Symbolic-Numeric Methods for Differential Equationsand Applications, NY, 2018

Satya S. Samal Tropical Geometry in Biology 20th July 2018 1 / 37

Outline

1 Motivation

2 Metastable Regimes

3 Tropical Geometry

4 Model Reduction

5 Symbolic Dynamics

6 Robustness Analysis

7 Challenges

8 Conclusion


Motivation

Tipping points / Critical transitions

Deviation of few system parameters qualitatively affect system behaviour.

Sudden change in a dynamical system’s state leading to bifurcations, phasetransitions,...

Changes could be predictive.


Motivation

Precision medicine

Predict therapy outcome (at individual/micro-segments).Extrapolation of mathematical models.

Heterogeneity of patients.Patient specificity parameters in models.

Non-stationary time series.Non constancy of underlying biological mechanism due to(clinical/biological) perturbations.

For example, alterations in signalling pathways (such asMAPK/PI3K).

Pathway redundancy and multiple feedback regulation areobstacles against cancer targeted therapies.


Motivation

Biological States

Biology is often understood as sequence of “biologicallyinterpretable states”.Such states can be thought of being slow regions.


Lobo, Neethan A., et al.(2007), Tyson, John J., et al.(2002)

Metastable Regimes

Low-Dimensional Sub-Manifold(s)

System of Ordinary Differential Equations (ODEs) often modelbiological processes e.g. metabolism, signalling.Many times, asymptotic behaviour of such systems evolve on alow-dimensional submanifold of the phase space (slow regions).

Maas, Ulrich et al.(1992), Chiavazzo, Eliodoro et al.(2007), Hung, Patrick et al.(2002)


Metastable Regimes

Metastable Regimes

Trajectories (of ODEs) consist of transitionsbetween slow regions.

Slow regions are denoted by low dimensionalsubmanifolds are called metastable states.

Metastable states may correspond to biologicallyobservable states (might even have names inbiological literature).

In our work, the metastable states correspond totropical equilibration (TE) solutions.

Slowness follows from thecompensation of dominant monomials.

Crazy-quilt to describe a patchy landscape ofmultiscale networks dynamics.


Tropical Geometry

Tropical Geometry: Basics

In tropical arithmetic, tropical addition (denoted by x ⊕ y = min(x , y)) andtropical multiplication (denoted by x � y = x + y ) of two numbers is theirminimum and sum in classical arithmetic.

The basic structure in tropical arithmetic is the tropical semiring which is a setdefined by (R ∪ {∞},⊕,�).

Tropical as limit of classical case: Let x and y be the powers of an auxiliaryvariable ε represented as εx and εy , where ε is a positive real number.

Tropical addition can be described as x ⊕ y = logε(εx + εy ) whichevaluates to min(x, y) if ε→ 0 and if ε→∞ it evaluates to max(x, y).Similarly, tropical multiplication can be described as x � y = logε(εxεy )which evaluates to x + y .


Tropical Geometry

Background: Tropical Geometry: Tropicalization

A tropical monomial is the tropical multiplication of these variables whererepetitions are allowed.

A Tropical polynomial is a piecewise linear concave function which is given asthe minimum of a finite set of linear functions with integer coefficients.

The tropicalization of f (x , y , ε) =∑

(i,j)∈A bij (ε)x iy j (whose coefficients arerational functions of a small parameter ε) denoted by T (f (x , y , ε)) ismin

(i,j)∈A(γij + ix + jy).

The tropical zeros are determined by computing the points at which the minimumof the tropical polynomial is attained at least twice. For example, consider anytwo points (i ′, j ′) and (i ′′, j ′′) in A, the computation of tropical zeros translates tosolving the following systems of linear inequalitiesγi′ j′ + i ′x + j ′y = γi′′ j′′ + i ′′x + j ′′y ≤ γij + ix + jy for (i, j) ∈ A where (i ′, j ′) and(i ′′, j ′′) range over the distinct points in A.

The set of tropical zeros (i.e. union of solution polytopes) of a tropicalpolynomial is called a tropical hypersurface.


Tropical Geometry

Tropical Geometry: Prevariety

A tropical prevariety is defined as the intersection of a finite number oftropical hypersurfaces, denoted by V (T (f1, f2, . . . , fk )) = ∩i∈[1,k ]T (fi )where T (f1, f2, . . . , fk ) and V (T (f1, f2, . . . , fk )) represent the set oftropicalization of the multivariate polynomials and the common tropicalzeros respectively.

A tropical variety is the intersection of all tropical hypersurfaces thatbelong to the ideal I generated by the polynomialsf1, f2, . . . , fk ,V (T (I)) = ∩f∈IT (f ) where T (I) represents the set oftropicalization of the elements of I and V (T (I)) denotes their commontropical zeros.

The tropical variety is within the tropical prevariety, but the reciprocalproperty is not always true.


Tropical Geometry

Tropical Geometry: Newton-Puiseux Series

Objective is to solve polynomials over the field of the Newton-Puiseux seriesdefined by C {{ε}}, where ε plays the role of indeterminate in the formal powerseries.

x(ε) = τ1εa1 + τ2ε

a2 + · · · , where τi ∈ C, and a1 < a2 < · · · are rationalnumbers with common denominator.

For an univariate polynomialf (x , ε) = Ad (ε)xd + Ad−1(ε)xd−1 + . . .+ A1(ε)x + A0(ε)

Puiseux theorem

The field of Puiseux series denoted by C {{ε}} is algebraically closed and thepolynomial f (x , ε) has d roots counting multiplicities, in the field of C {{ε}}.

The roots of f (x , ε) are x(ε) = xεa1 + higher order terms in ε. The possiblevalues of a1 with the lowest order terms as shown below

Ad xd1 εγd +da1 + Ad−1xd−1εγd +(d−1)a1 + . . .+ A1x1εγ1+a1 + A0ε

γ0 = 0

The possible values of a1 are solutions ofmin(γd + da1, γd−1 + (d − 1)a1, . . . , γ1 + a1, γ0) where the min is attained at leasttwice.


Tropical Geometry

Reaction Network to ODE

ODE system obtained from biochemical reaction network assumingmass-action kinetics

dxi

dt=

∑j

kjSijxαj , 1 ≤ i ≤ n

where kj > 0 are kinetic constants, Sij are the entries of thestoichiometric matrix. xi , i ∈ [1,n] are the species concentrations, nbeing the number of speciesGiven reaction A + B → C of kinetic constant k and satisfying the massaction law, has S11 = −1,S21 = −1,S31 = 1, α1 = (1,1,0), whichcorrespond to the kinetic equations

dx1

dt= −kx1x2,

dx2

dt= −kx1x2,

dx3

dt= kx1x2, (1)

where x1, x2, x3 are the concentrations of A, B, C, respectively.


Tropical Geometry

Tropical Equilibration Problem

Rescaling of ODE systemParameters of the ODE can be written as kj = kjε

γj , whereexponent γj = round(log(kj )/ log(ε)) and the variables as xi = xiε

ai .The rescaled ODE system dxi

dt =∑

j εµj−ai kjSij x

αj ,whereµj (a) = γj + 〈a, αj〉, and 〈, 〉 stands for the dot product in Rn.

Extracting the exponent vector.TE problem involves computing the dominant monomials based on theexponent vector i.e. finding a vector a such that:minj,Sij>0(γj + 〈a, αj〉) = minj,Sij<0(γj + 〈a, αj〉)Thus non-linear polynomials are replaced with piecewise linearfunctions.Tropical geometry (through Newton polytopes) is an algebraic method toaddress such a problem.Solving system of polynomial equations in tropical semi-ring(R ∪ {∞},+, x).

Relationship to classical polynomials by valuation theory (i.e.Puiseux series solutions).Computing the tropical prevariety.


Tropical Geometry

Computation of Tropical Equilibrations

Equation−x6

1 + x31 x2− x3

1 + x1x22

Order of the variablesx1 = x1ε

a1 , x2 = x2εy

Order of the monomialsx6

1 = x1ε6a1

x31 x2 = x1ε

3a1 x2εa2

x31 = x1ε

3a1

x1x22 = x1ε

a1 x2ε2a2

All the monomial coefficients have orderzero in ε and we want to solve the tropicalproblemmin(3a1 + a2, a1 + 2a2) = min(6a1, 3a1).

The thick edges satisfy the sign condition,whereas the dashed edge does not satisfythis condition.

Branches of tropical solutions correspond to halflines (orthogonal to the thick edges of newtonpolytope) and are given by{a1 = a2 ≥ 0},{a1 ≤ 0, a2 = 5/2a1}.


Tropical Geometry

Minimal and Connected branches

Branches: Equivalence classes of TE solutions.For each branch there exists a unique convex polytope.

Minimal branch: Branch corresponding to maximal polytope withrespect to inclusion.

A branch B with a convex polytope Pi is minimal if Pi ⊂ P ′i for all iwhere P ′i is the convex polytope for branch B′ implies B′ = B or B′

is empty.Connected branches: Checking the intersection between twobranches.

Checking the intersection between two convex polytopes Pi and Pj(corresponding to minimal branches Mi and Mj ) if whether Pi ∩ Pj isnon void for all i 6= j .


Model Reduction

Slow/fast Systems: Tikhonov Theorem

After time rescaling, the differential equations describing the dynamicsof a system with fast variables x and slow variables y read as:

dxdt

= 1ηf(x, y),

dydt

= g(x, y).

where η is a fast/slow timescale ratio.Tikhonov: If for any y the fast dynamics has a hyperbolic pointattractor, then after a quick transition the system evolves according to:dydt = g(x, y) and f (x , y) = 0 fast variables are slaved by slow ones.


Model Reduction

Model Reduction Steps

Determine the slow/fast decomposition (who are the small parameter η, the slowand the fast variables?): Jacobian based numerical methods (CSP, ILDM,COPASI implementation); tropical geometry based methods.

Solve f (x , y) = 0 for x (fast variables): hard, few symbolic methods(sparsepolynomial systems ?).

Pool reactions (elementary modes) / prune species (conservation laws).


Model Reduction

Michaelis-Menten Enzymatic Kinetics

The irreversible Michaelis-Menten kinetics consist of threereactions:

S + Ek1k−1

ESk2→ P + E ,

where S,ES,E ,P represent the substrate, theenzyme-substrate complex, the enzyme and the product,respectively. The corresponding ODEs are:

x1 = −k1x1x3 + k−1x2,

x2 = k1x1x3 − (k−1 + k2)x2,

x3 = −k1x1x3 + (k−1 + k2)x2,

x4 = k2x2

where x1 = [S], x2 = [ES], x3 = [E ], x4 = [P].The system has two conservation laws x2 + x3 = e0 andx1 + x2 + x4 = s0. The values e0 and s0 of the conservationlaws result from the the initial conditions, namelye0 = x2(0) + x3(0) and s0 = x1(0) + x2(0) + x4(0).


Model Reduction

Model Reduction: Michaelis-Menten Enzymatic Kinetics

The reduced model after eliminating variables x3 and x4 using conservation laws (e0

and s0):

x1 = −k1x1(e0 − x2) + k−1x2,

x2 = k1x1(e0 − x2)− (k−1 + k2)x2.

For Quasi-equilibrium (γ−1 < γ2), the corresponding tropical solutions are:

a2 = γe, a1 ≤ γm (saturation regime)

a2 = a1 + γe − γm, a1 ≥ γm (linear regime)

where γm = γ−1 − γ1 (order of the parameter Km = k−1/k1).For linear regime, the fast truncated system (removing the dominant monomials)reads:

x1 = −k1x1e0 + k−1x2,

x2 = k1x1e0 − k−1x2.

Introduce new (slow) variable y = x1 + x2.

y = −(Vmax/Km)x1, where Vmax = k2e0

Likewise, for saturated regime: y = −Vmax .Satya S. Samal Tropical Geometry in Biology 20th July 2018 19 / 37

Model Reduction

Model Reduction Results: Cell Cycle

x1 x2x6 x5

x3 x4

k4 x

2 x5k 1

x 3

k2x1

k3x2

k9x4x23

k10x4

k8 x

6 k 6

Full model

x3 x4

yx6k 1

x 3

k9x4x23

k10x4

k8 x

6

k6

k6

Reduced model


Symbolic Dynamics

Symbolic Dynamics

A trajectory is a sequence of minimal branches.The minimal branches are alphabets and a trajectory is thesequence of alphabets resulting in symbolic dynamics.Example: For minimal branches 1,2,3 an example of finite statemachine generating symbolic dynamics.


Symbolic Dynamics

Problem Statement

InputSystem of ODEs with polynomial vector field (described by massaction kinetics)Fixed kinetic parameters of the model.

OutputMinimal branches corresponding to metastable states.Stochastic finite state automaton (with transition probabilities).


Symbolic Dynamics

Finite State Automaton

StatesMinimal branches (or metastable states).

TransitionsIntersections between minimal branches (connectivity graph).

Weights or Probabilities between statesTrajectories of ODE were simulated and their distance to minimalbranches is computed.

States of automata are constructed independent of initialconditions of ODE system.


Symbolic Dynamics

Results: Maximal Polytopes

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Number of equations

Min

imal

bra

nche

s

1 9 17 25 33 41

1

3

5

7

9

11

13

15

17

Figure: Minimal branches against number of equations in the model. Models obtained fromBiomodels database.


Symbolic Dynamics

MAPK cascade : Huang and Ferrell 96


Symbolic Dynamics

Symbolic dynamics of MAPK cascade

B1

B2

B3

B4

B5

B6

B7

B8

B9

B10


Symbolic Dynamics

Symbolic dynamics of MAPK cascade

B1

B2

B3

B4

B5

B6

B7

B8

B9

B10

1.0

1.0

1.0

1.0

1.0

1.0

1.0

1.0

0.98

0.02

1.0


Symbolic Dynamics

Continuous trajectory


Symbolic Dynamics

Discrete sequence of branches


Robustness Analysis

Sensitive/Robust Parameters

Compute nominal minimal branches with the given model parameters.Thereafter, the perturbed tropical solution sets by perturbing parameter ordersare γj − 3, γj − 2, γj − 1, γj + 1, γj + 2, γj + 3 respectively.

Compute the distance as Dji = mint∈T{min

t′∈T′ ji{||t − t ′||}} where T and T

′ ji

denote the sets of representative point(s) sampled from the polytopes in M ∩ Γand M

′ ji ∩ Γ respectively. || · || is the Lp norm distance metric.

Tk = RepresentativePoint(Mk ∩ Γ ), k = 1 . . . p whereΓ = {a ∈ Rn|lb ≤ ai ≤ ub, i = 1 . . . n}.

Parameter sensitivity of j th parameter: Dj = 1l

l∑i=1

Dji .

Parameter sensitivity projection on variable Xm: Dj |Xm = 1l

l∑i=1

Dji |Xm.


Robustness Analysis

Network Model

The model was motivated by experimentalworks on the Heregulin stimulated ErbBreceptor and demonstrates the Akt-inducedinhibition of the MAPK pathway viaphosphorylation of Raf-1 (Hatakeyama, M. etal (2013), Biomodels ID: BIOMD0000000146).

This CRN model has 33 species and 34reactions. 21 reactions haveMichaelis-Menten kinetics and 12 have massaction kinetic laws.


Robustness Analysis

Results: Distances

Parameter sensitivities are provided as normalized average distancesin 3 versions: D1 full distance, D2 distance along MAPKPP axis, D3distance along AKTPiPP axis. p = 2 in Lp norm.

Par Protein D1 D2 D3k77 DUSP 0.584 0.610 0k78 PP2A 0.953 0.943 1.000k79 AKT3 0.544 0 0.916k81 PI3K 0.629 0 0.124k82 MAPK 0.342 1.000 0k83 MEK 0.594 0.971 0k84 RAF 0.405 0.388 0k85 RAS 0.258 0 0k86 SOS/GRB2 0.945 0.722 0k87 SHC 1 0.943 0k89 EGFR 0.903 0 0.245

Table of parameter sensitivities and histogram for D1.


Robustness Analysis

Change in Protein Concentrations

Par P-values BRCA P-values SKCMk77 0.355 0.178k79 0.004 0.068k81 0.037 0.396k82 0.006 0.0121k83 1.852e-06 0.068k84 6.84e-09 0.012k85 6.848e-09 0.696k87 1.092e-06 0.702k89 2.306e-07 0.068

Adjusted P-values of BRCA (normal versus primary samples)and SKCM (metastatic versus primary samples) cancers fromTCPA protein database.

Quantify the overlap withtropical sensitivity scoresArea under the curve (AUC):

Breast cancer data:0.55Skin Cutaneous Melanoma:0.85Breast cancer subtypes:(average): 0.72*0.50 is random guessing


Challenges

Challenges

The number of minimal branches not necessarily scales withnumber of variables.Solve the inverse problem i.e. learn the orders of parameters fromdata i.e. (approximate) tropical interpolation.Model reduction for ODEs with sums of fractionsAdditional constraints (e.g. stability constraints) to the tropicalequilibration branchesTipping points and dynamical regimes of large biological networkssymbolically i.e. scalability.


Conclusion

Conclusion

Describing the dynamics with metastable regimes.Tropical geometry to determine slow-fast variables for model orderreduction.A method to transform the continuous dynamics to discrete(through finite state automaton).

Questions like reachability can be answered.Metastable states may correspond to biologically observablestates.

Transition probabilities explain the interplay between them (coarsegrain properties of ensemble of trajectories).

Global method to assess robustness of biological networks.“The purpose of computing is insight, not numbers” by R.Hamming


Conclusion

AcknowledgementsPeople

Prof. Dr. Ovidiu Radulescu

Prof. Dr. Andreas Weber

Prof. Dr. Dima Grigoriev

Dr. Holger Fröhlich

Prof. Dr. Andreas Schuppert

Christoph Lüders

Jeyashree Krishnan

Ali Esfahani

Funding

This work has been partiallysupported by the bilateral projectANR-17-CE40-0036 andDFG-391322026 SYMBIONT.

Fellowship from ComputationalSciences and Engineering profilearea, RWTH Aachen University.


Conclusion

References

S. S. Samal, D. Grigoriev, H. Fröhlich, A. Weber, and O. Radulescu, A geometricmethod for model reduction of biochemical networks with polynomial ratefunctions. Bulletin of Mathematical Biology, 77(12): 2180-2211, 2015.

S. S. Samal, A. Naldi, D. Grigoriev, A. Weber, N. Théret, and O. Radulescu.Geometric analysis of pathways dynamics: application to versatility of TGF-βreceptors. Biosystems, 149: 3-14, 2016.

C. Lüders, S. S. Samal O. Radulescu, A. Weber, PtCut: A Program to CalculateTropical Equilibria (http://wrogn.com).

S. S. Samal, J. Krishnan, A. H. Esfahani, C. Lüders, A. Weber, O. Radulescu,Metastable regimes and tipping points of biochemical networks with potentialapplications in precision medicine (Manuscript submitted).


http://wrogn.com

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