Analysis of Biochemical Reaction Network Systems Using Tropical Geometry

Transcription

1 Analysis of Biochemical Reaction Network Systems Using Tropical Geometry Satya Swarup Samal Joint Research Center for Computational Biomedicine (JRC-COMBINE) RWTH Aachen University Workshop on Symbolic-Numeric Methods for Differential Equations and Applications, NY, 2018 Satya S. Samal Tropical Geometry in Biology 20th July / 37

2 Outline 1 Motivation 2 Metastable Regimes 3 Tropical Geometry 4 Model Reduction 5 Symbolic Dynamics 6 Robustness Analysis 7 Challenges 8 Conclusion Satya S. Samal Tropical Geometry in Biology 20th July / 37

3 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, phase transitions,... Changes could be predictive. Satya S. Samal Tropical Geometry in Biology 20th July / 37

4 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 as MAPK/PI3K). Pathway redundancy and multiple feedback regulation are obstacles against cancer targeted therapies. Satya S. Samal Tropical Geometry in Biology 20th July / 37

5 Motivation Biological States Biology is often understood as sequence of biologically interpretable states. Such states can be thought of being slow regions. Lobo, Neethan A., et al.(2007), Tyson, John J., et al.(2002) Satya S. Samal Tropical Geometry in Biology 20th July / 37

6 Metastable Regimes Low-Dimensional Sub-Manifold(s) System of Ordinary Differential Equations (ODEs) often model biological processes e.g. metabolism, signalling. Many times, asymptotic behaviour of such systems evolve on a low-dimensional submanifold of the phase space (slow regions). Maas, Ulrich et al.(1992), Chiavazzo, Eliodoro et al.(2007), Hung, Patrick et al.(2002) Satya S. Samal Tropical Geometry in Biology 20th July / 37

7 Metastable Regimes Metastable Regimes Trajectories (of ODEs) consist of transitions between slow regions. Slow regions are denoted by low dimensional submanifolds are called metastable states. Metastable states may correspond to biologically observable states (might even have names in biological literature). In our work, the metastable states correspond to tropical equilibration (TE) solutions. Slowness follows from the compensation of dominant monomials. Crazy-quilt to describe a patchy landscape of multiscale networks dynamics. Satya S. Samal Tropical Geometry in Biology 20th July / 37

8 Tropical Geometry: Basics Tropical Geometry In tropical arithmetic, tropical addition (denoted by x y = min(x, y)) and tropical multiplication (denoted by x y = x + y) of two numbers is their minimum and sum in classical arithmetic. The basic structure in tropical arithmetic is the tropical semiring which is a set defined by (R { },, ). Tropical as limit of classical case: Let x and y be the powers of an auxiliary variable ε represented as ε x and ε y, where ε is a positive real number. Tropical addition can be described as x y = log ε (ε x + ε y ) which evaluates 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 = log ε (ε x ε y ) which evaluates to x + y. Satya S. Samal Tropical Geometry in Biology 20th July / 37

9 Tropical Geometry Background: Tropical Geometry: Tropicalization A tropical monomial is the tropical multiplication of these variables where repetitions are allowed. A Tropical polynomial is a piecewise linear concave function which is given as the minimum of a finite set of linear functions with integer coefficients. The tropicalization of f (x, y, ε) = (i,j) A b ij(ε)x i y j (whose coefficients are rational functions of a small parameter ε) denoted by T (f (x, y, ε)) is (γ ij + ix + jy). min (i,j) A The tropical zeros are determined by computing the points at which the minimum of the tropical polynomial is attained at least twice. For example, consider any two points (i, j ) and (i, j ) in A, the computation of tropical zeros translates to solving the following systems of linear inequalities γ i j + i x + j y = γ i j + i x + j y γ ij + 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 tropical polynomial is called a tropical hypersurface. Satya S. Samal Tropical Geometry in Biology 20th July / 37

10 Tropical Geometry Tropical Geometry: Prevariety A tropical prevariety is defined as the intersection of a finite number of tropical hypersurfaces, denoted by V (T (f 1, f 2,..., f k )) = i [1,k] T (f i ) where T (f 1, f 2,..., f k ) and V (T (f 1, f 2,..., f k )) represent the set of tropicalization of the multivariate polynomials and the common tropical zeros respectively. A tropical variety is the intersection of all tropical hypersurfaces that belong to the ideal I generated by the polynomials f 1, f 2,..., f k, V (T (I)) = f I T (f ) where T (I) represents the set of tropicalization of the elements of I and V (T (I)) denotes their common tropical zeros. The tropical variety is within the tropical prevariety, but the reciprocal property is not always true. Satya S. Samal Tropical Geometry in Biology 20th July / 37

11 Tropical Geometry Tropical Geometry: Newton-Puiseux Series Objective is to solve polynomials over the field of the Newton-Puiseux series defined by C {{ε}}, where ε plays the role of indeterminate in the formal power series. x(ε) = τ 1 ε a 1 + τ 2 ε a 2 +, where τ i C, and a 1 < a 2 < are rational numbers with common denominator. For an univariate polynomial f (x, ε) = A d (ε)x d + A d 1 (ε)x d A 1 (ε)x + A 0 (ε) Puiseux theorem The field of Puiseux series denoted by C {{ε}} is algebraically closed and the polynomial f (x, ε) has d roots counting multiplicities, in the field of C {{ε}}. The roots of f (x, ε) are x(ε) = xε a 1 + higher order terms in ε. The possible values of a 1 with the lowest order terms as shown below Ā d x d 1 ε γ d +da 1 + Ād 1 x d 1 ε γ d +(d 1)a Ā1 x 1 ε γ 1+a 1 + Ā0ε γ 0 = 0 The possible values of a 1 are solutions of min(γ d + da 1, γ d 1 + (d 1)a 1,..., γ 1 + a 1, γ 0 ) where the min is attained at least twice. Satya S. Samal Tropical Geometry in Biology 20th July / 37

12 Reaction Network to ODE Tropical Geometry ODE system obtained from biochemical reaction network assuming mass-action kinetics dx i = k j S ij x αj, 1 i n dt j where k j > 0 are kinetic constants, S ij are the entries of the stoichiometric matrix. x i, i [1, n] are the species concentrations, n being the number of species Given reaction A + B C of kinetic constant k and satisfying the mass action law, has S 11 = 1, S 21 = 1, S 31 = 1, α 1 = (1, 1, 0), which correspond to the kinetic equations dx 1 dt = kx 1 x 2, dx 2 dt = kx 1 x 2, dx 3 dt = kx 1 x 2, (1) where x 1, x 2, x 3 are the concentrations of A, B, C, respectively. Satya S. Samal Tropical Geometry in Biology 20th July / 37

13 Tropical Geometry Tropical Equilibration Problem Rescaling of ODE system Parameters of the ODE can be written as k j = k j ε γ j, where exponent γ j = round(log(k j )/ log(ε)) and the variables as x i = x i ε a i. The rescaled ODE system d x i dt = j εµ j a i kj S ij x αj, whereµ j (a) = γ j + a, α j, and, stands for the dot product in R n. Extracting the exponent vector. TE problem involves computing the dominant monomials based on the exponent vector i.e. finding a vector a such that: min j,sij >0(γ j + a, α j ) = min j,sij <0(γ j + a, α j ) Thus non-linear polynomials are replaced with piecewise linear functions. Tropical geometry (through Newton polytopes) is an algebraic method to address 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. Satya S. Samal Tropical Geometry in Biology 20th July / 37

14 Tropical Geometry Computation of Tropical Equilibrations Equation x x3 1 x 2 x x 1x 2 2 Order of the variables x 1 = x 1 ε a 1, x 2 = x 2 ε y Order of the monomials x1 6 = x 1ε 6a 1 x1 3 x 2 = x 1 ε 3a 1 x 2 ε a 2 x1 3 = x 1ε 3a 1 x 1 x2 2 = x 1ε a 1 x 2 ε 2a 2 All the monomial coefficients have order zero in ɛ and we want to solve the tropical problem min(3a 1 + a 2, a 1 + 2a 2 ) = min(6a 1, 3a 1 ). The thick edges satisfy the sign condition, whereas the dashed edge does not satisfy this condition. Branches of tropical solutions correspond to half lines (orthogonal to the thick edges of newton polytope) and are given by {a 1 = a 2 0}, {a 1 0, a 2 = 5/2a 1 }. Satya S. Samal Tropical Geometry in Biology 20th July / 37

15 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 with respect to inclusion. A branch B with a convex polytope P i is minimal if P i P i for all i where P i is the convex polytope for branch B implies B = B or B is empty. Connected branches: Checking the intersection between two branches. Checking the intersection between two convex polytopes P i and P j (corresponding to minimal branches M i and M j ) if whether P i P j is non void for all i j. Satya S. Samal Tropical Geometry in Biology 20th July / 37

16 Model Reduction Slow/fast Systems: Tikhonov Theorem After time rescaling, the differential equations describing the dynamics of a system with fast variables x and slow variables y read as: dx dt dy dt = 1 η f(x, y), = g(x, y). where η is a fast/slow timescale ratio. Tikhonov: If for any y the fast dynamics has a hyperbolic point attractor, then after a quick transition the system evolves according to: dy dt = g(x, y) and f (x, y) = 0 fast variables are slaved by slow ones. Satya S. Samal Tropical Geometry in Biology 20th July / 37

17 Model Reduction Steps Model Reduction Determine the slow/fast decomposition (who are the small parameter η, the slow and 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(sparse polynomial systems?). Pool reactions (elementary modes) / prune species (conservation laws). Satya S. Samal Tropical Geometry in Biology 20th July / 37

18 Model Reduction Michaelis-Menten Enzymatic Kinetics The irreversible Michaelis-Menten kinetics consist of three reactions: S + E k 1 ES k 2 P + E, k 1 where S, ES, E, P represent the substrate, the enzyme-substrate complex, the enzyme and the product, respectively. The corresponding ODEs are: ẋ 1 = k 1 x 1 x 3 + k 1 x 2, ẋ 2 = k 1 x 1 x 3 (k 1 + k 2 )x 2, ẋ 3 = k 1 x 1 x 3 + (k 1 + k 2 )x 2, ẋ 4 = k 2 x 2 where x 1 = [S], x 2 = [ES], x 3 = [E], x 4 = [P]. The system has two conservation laws x 2 + x 3 = e 0 and x 1 + x 2 + x 4 = s 0. The values e 0 and s 0 of the conservation laws result from the the initial conditions, namely e 0 = x 2 (0) + x 3 (0) and s 0 = x 1 (0) + x 2 (0) + x 4 (0). Satya S. Samal Tropical Geometry in Biology 20th July / 37

19 Model Reduction Model Reduction: Michaelis-Menten Enzymatic Kinetics The reduced model after eliminating variables x 3 and x 4 using conservation laws (e 0 and s 0 ): ẋ 1 = k 1 x 1 (e 0 x 2 ) + k 1 x 2, ẋ 2 = k 1 x 1 (e 0 x 2 ) (k 1 + k 2 )x 2. For Quasi-equilibrium (γ 1 < γ 2 ), the corresponding tropical solutions are: a 2 = γ e, a 1 γ m (saturation regime) a 2 = a 1 + γ e γ m, a 1 γ m (linear regime) where γ m = γ 1 γ 1 (order of the parameter K m = k 1 /k 1 ). For linear regime, the fast truncated system (removing the dominant monomials) reads: ẋ 1 = k 1 x 1 e 0 + k 1 x 2, ẋ 2 = k 1 x 1 e 0 k 1 x 2. Introduce new (slow) variable y = x 1 + x 2. ẏ = (V max/k m)x 1, where V max = k 2 e 0 Likewise, for saturated regime: ẏ = V max. Satya S. Samal Tropical Geometry in Biology 20th July / 37

20 Model Reduction Model Reduction Results: Cell Cycle x6 x1 k 2 x 1 k 3 x 2 x2 x5 k1x3 k4x2x5 k8x6 k6 x3 k 10 x 4 x4 k 9 x 4 x 2 3 x6 Full model y k 6 k8x6 k1x3 x3 k 10 x 4 k6 x4 k 9 x 4 x 2 3 Reduced model Satya S. Samal Tropical Geometry in Biology 20th July / 37

21 Symbolic Dynamics Symbolic Dynamics A trajectory is a sequence of minimal branches. The minimal branches are alphabets and a trajectory is the sequence of alphabets resulting in symbolic dynamics. Example: For minimal branches 1, 2, 3 an example of finite state machine generating symbolic dynamics. Satya S. Samal Tropical Geometry in Biology 20th July / 37

22 Symbolic Dynamics Problem Statement Input Output System of ODEs with polynomial vector field (described by mass action kinetics) Fixed kinetic parameters of the model. Minimal branches corresponding to metastable states. Stochastic finite state automaton (with transition probabilities). Satya S. Samal Tropical Geometry in Biology 20th July / 37

23 Finite State Automaton Symbolic Dynamics States Minimal branches (or metastable states). Transitions Intersections between minimal branches (connectivity graph). Weights or Probabilities between states Trajectories of ODE were simulated and their distance to minimal branches is computed. States of automata are constructed independent of initial conditions of ODE system. Satya S. Samal Tropical Geometry in Biology 20th July / 37

24 1 9 Symbolic Dynamics Results: Maximal Polytopes Minimal branches Number of equations Figure: Minimal branches against number of equations in the model. Models obtained from Biomodels database. Satya S. Samal Tropical Geometry in Biology 20th July / 37

25 Symbolic Dynamics MAPK cascade : Huang and Ferrell 96 Satya S. Samal Tropical Geometry in Biology 20th July / 37

26 Symbolic Dynamics Symbolic dynamics of MAPK cascade B9 B7 B10 B8 B6 B3 B1 B4 B5 B2 Satya S. Samal Tropical Geometry in Biology 20th July / 37

27 1.0 Symbolic Dynamics Symbolic dynamics of MAPK cascade B9 1.0 B7 B B8 B6 B B1 B B5 1.0 B2 Satya S. Samal Tropical Geometry in Biology 20th July / 37

28 Symbolic Dynamics Continuous trajectory Satya S. Samal Tropical Geometry in Biology 20th July / 37

29 Symbolic Dynamics Discrete sequence of branches Satya S. Samal Tropical Geometry in Biology 20th July / 37

30 Robustness Analysis Sensitive/Robust Parameters Compute nominal minimal branches with the given model parameters. Thereafter, the perturbed tropical solution sets by perturbing parameter orders are γ j 3, γ j 2, γ j 1, γ j + 1, γ j + 2, γ j + 3 respectively. Compute the distance as D j i = min t T {min t T j { t t }} where T and T j i i denote the sets of representative point(s) sampled from the polytopes in M Γ and M j i Γ respectively. is the L p norm distance metric. T k = RepresentativePoint(M k Γ ), k = 1... p where Γ = {a R n lb a i ub, i = 1... n}. l Parameter sensitivity of j th parameter: D j = 1 D j l i. Parameter sensitivity projection on variable X m: D j X m = 1 l i=1 l D j i Xm. i=1 Satya S. Samal Tropical Geometry in Biology 20th July / 37

31 Robustness Analysis Network Model The model was motivated by experimental works on the Heregulin stimulated ErbB receptor and demonstrates the Akt-induced inhibition of the MAPK pathway via phosphorylation of Raf-1 (Hatakeyama, M. et al (2013), Biomodels ID: BIOMD ). This CRN model has 33 species and 34 reactions. 21 reactions have Michaelis-Menten kinetics and 12 have mass action kinetic laws. Satya S. Samal Tropical Geometry in Biology 20th July / 37

32 Robustness Analysis Results: Distances Parameter sensitivities are provided as normalized average distances in 3 versions: D 1 full distance, D 2 distance along MAPKPP axis, D 3 distance along AKTPiPP axis. p = 2 in L p norm. Par Protein D 1 D 2 D 3 k 77 DUSP k 78 PP2A k 79 AKT k 81 PI3K k 82 MAPK k 83 MEK k 84 RAF k 85 RAS k 86 SOS/GRB k 87 SHC k 89 EGFR Table of parameter sensitivities and histogram for D 1. Satya S. Samal Tropical Geometry in Biology 20th July / 37

33 Robustness Analysis Change in Protein Concentrations Par P-values BRCA P-values SKCM k k k k k e k e k e k e k e Adjusted P-values of BRCA (normal versus primary samples) and SKCM (metastatic versus primary samples) cancers from TCPA protein database. Quantify the overlap with tropical sensitivity scores Area under the curve (AUC): Breast cancer data:0.55 Skin Cutaneous Melanoma: 0.85 Breast cancer subtypes: (average): 0.72 *0.50 is random guessing Satya S. Samal Tropical Geometry in Biology 20th July / 37

34 Challenges Challenges The number of minimal branches not necessarily scales with number of variables. Solve the inverse problem i.e. learn the orders of parameters from data i.e. (approximate) tropical interpolation. Model reduction for ODEs with sums of fractions Additional constraints (e.g. stability constraints) to the tropical equilibration branches Tipping points and dynamical regimes of large biological networks symbolically i.e. scalability. Satya S. Samal Tropical Geometry in Biology 20th July / 37

35 Conclusion Conclusion Describing the dynamics with metastable regimes. Tropical geometry to determine slow-fast variables for model order reduction. 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 observable states. Transition probabilities explain the interplay between them (coarse grain properties of ensemble of trajectories). Global method to assess robustness of biological networks. The purpose of computing is insight, not numbers by R. Hamming Satya S. Samal Tropical Geometry in Biology 20th July / 37

36 Conclusion Acknowledgements People 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 partially supported by the bilateral project ANR-17-CE and DFG SYMBIONT. Fellowship from Computational Sciences and Engineering profile area, RWTH Aachen University. Satya S. Samal Tropical Geometry in Biology 20th July / 37

37 Conclusion References S. S. Samal, D. Grigoriev, H. Fröhlich, A. Weber, and O. Radulescu, A geometric method for model reduction of biochemical networks with polynomial rate functions. Bulletin of Mathematical Biology, 77(12): , S. S. Samal, A. Naldi, D. Grigoriev, A. Weber, N. Théret, and O. Radulescu. Geometric analysis of pathways dynamics: application to versatility of TGF-β receptors. Biosystems, 149: 3-14, C. Lüders, S. S. Samal O. Radulescu, A. Weber, PtCut: A Program to Calculate Tropical Equilibria ( S. S. Samal, J. Krishnan, A. H. Esfahani, C. Lüders, A. Weber, O. Radulescu, Metastable regimes and tipping points of biochemical networks with potential applications in precision medicine (Manuscript submitted). Satya S. Samal Tropical Geometry in Biology 20th July / 37