Linear robustness analysis of non-linear biomolecular networks with kinetic perturbations Steffen Waldherr and Frank Allgöwer Institute for Systems Theory and Automatic Control Universität Stuttgart MSC 2011, Workshop on Robustness in Biological Systems
Robustness analysis Informal definition of robustness Robustness is a property that allows a system to maintain its function in the presence of internal and external perturbations. H. Kitano, 2004 System / Model: ˙ x = Sv ( x ) x ∈ R n : amounts of signalling molecules v ( x ) ∈ R m : reaction rates S ∈ R n × m : stoichiometric matrix Perturbations: kinetic perturbations (changes in reaction kinetics v ) Function: Qualitative dynamical behaviour (stability, oscillations, ...) Robustness analysis with kinetic perturbations, S. Waldherr 1 / 26
Formal robustness measures Smallest non-robust perturbation in ∞ - norm robust perturbations nominal point loss of function Non-robust perturbation = any deviation from the nominal point for which the system looses functionality Robustness measure = smallest norm of non-robust perturbations Robustness analysis with kinetic perturbations, S. Waldherr 2 / 26
Why robustness analysis? Robustness analysis for model evaluation Biological systems are robust ↔ models should be robust! Find parts of the system which need to be modelled carefully Evaluate effects of uncertainty on model predictions Robustness analysis for system understanding Understand mechanisms which confer robustness to biological systems Detect fragile points, e.g. for medical intervention Guide system design in bioengineering and synthetic biology Robustness analysis with kinetic perturbations, S. Waldherr 3 / 26
Outline Theory of kinetic perturbations 1 Robustness analysis with kinetic perturbations 2 Adaptation with kinetic perturbations 3 Robustness analysis with kinetic perturbations, S. Waldherr 4 / 26
Outline Theory of kinetic perturbations 1 Definition of kinetic perturbations Mapping kinetic perturbations to parameter variations Robustness analysis with kinetic perturbations 2 Adaptation with kinetic perturbations 3 Robustness analysis with kinetic perturbations, S. Waldherr 5 / 26
Kinetic perturbations – the basic idea Kinetic perturbations = reaction rate perturbations nominal rates v ( x ) to perturbed rates ˜ v ( x ) ⇒ nominal model ˙ x = Sv ( x ) to perturbed model ˙ x = S ˜ v ( x ) Consider nominal steady state x 0 : Sv ( x 0 ) = 0 Kinetic perturbation, if ˜ v ( x 0 ) = v ( x 0 ) Reaction rates in steady state unperturbed, but reaction rate slopes may vary: ˜ v ( x ) , increased slope v ( x ) v ( x ) = 0 . 7 x 1 v ( x ) , reduced slope ˜ x 0 = 1 x 1 2 Robustness analysis with kinetic perturbations, S. Waldherr 6 / 26
Formal definition of kinetic perturbations Definition The network ˙ x = S ˜ v ( x ) is said to be a kinetic perturbation of the nominal x = Sv ( x ) at x 0 ∈ R n with Sv ( x 0 ) = 0, if ˜ network ˙ v ( x 0 ) = v ( x 0 ) =: v 0 . Consider the reaction rate Jacobian V = ∂ v ∂ x For the perturbed network, we have V ( x 0 ) = ∂ ˜ v ˜ ∂ x ( x 0 ) = V ( x 0 ) + ¯ ∆ Local effects of the kinetic perturbation are determined by ¯ ∆ x 0 ,ℓ v 0 , i ¯ Scaled perturbation ∆ i ℓ = ∆ i ℓ � ∆ � is a measure for the perturbation size Steady state concentration x 0 and flux v 0 are unperturbed Steady state reaction slopes V ( x 0 ) are perturbed by ¯ ∆ Robustness analysis with kinetic perturbations, S. Waldherr 7 / 26
Mapping kinetic perturbations to parameter changes Considering generalised mass action (GMA) networks � n ℓ = 1 x α i ℓ GMA reaction rate: v i ( x ) = k i ℓ α i ℓ > 0: x ℓ is a substrate for / activates v i α i ℓ < 0: x ℓ inhibits v i Non-integer kinetic orders α i ℓ Effect of constrained diffusion (fractal reaction kinetics) Result of model simplifications (as in S-systems, Savageau et al. ) v i ( x ) = ˜ � n ℓ = 1 x ˜ α i ℓ Perturbed reaction rate ˜ k i ℓ Parameter change Kinetic perturbation α i ℓ = α i ℓ + ∆ i ℓ ˜ ⇔ ∂ x ℓ + v 0 , i ∂ ˜ v i ∂ v i n ∂ x ℓ = x 0 ,ℓ ∆ i ℓ ˜ � x − ∆ i ℓ k i = k i 0 ,ℓ ℓ = 1 Robustness analysis with kinetic perturbations, S. Waldherr 8 / 26
Biochemical interpretation of kinetic perturbations Are kinetic perturbations biochemically plausible? Yes, because kinetic perturbations map to parameter variations in reasonable model classes for intracellular networks such as GMA networks enzymatic networks (Michaelis-Menten or Hill reaction kinetics) genetic networks (sigmoidal activation & inhibition functions) Biochemical interpretation Positive ∆ i ℓ : more cooperativity in reaction i with respect to species ℓ Negative ∆ i ℓ : increase saturation in reaction i with respect to species ℓ Robustness analysis with kinetic perturbations, S. Waldherr 9 / 26
Outline Theory of kinetic perturbations 1 Robustness analysis with kinetic perturbations 2 Kinetic perturbations affect dynamical properties Robustness analysis via the structured singular value Example: Oscillations in the MAPK cascade Adaptation with kinetic perturbations 3 Robustness analysis with kinetic perturbations, S. Waldherr 10 / 26
Inducing bifurcations by kinetic perturbations Example “network” kx 2 ˙ v 1 = x = v 1 − v 2 1 + Mx 2 X v 2 = x v 1 v 2 v 2 v ( x ) ∂ ˜ ∂ x ( 1 ) + ¯ ∂ x ( 1 ) = ∂ v 1 v 1 ∆ v 1 1 ¯ ∆ = 0: x 0 stable x 0 = 1 ∆ = ¯ ¯ ∆ ∗ : transcritical bifurcation at x 0 ∆ > ¯ ¯ ∆ ∗ : x 0 unstable x 1 2 Loss of bistability by a kinetic perturbation Robustness analysis with kinetic perturbations, S. Waldherr 11 / 26
Inducing bifurcations by kinetic perturbations Example “network” kx 2 ˙ v 1 = x = v 1 − v 2 1 + Mx 2 X v 2 = x v 1 v 2 v 2 v ( x ) ˜ v 1 ∂ ˜ ∂ x ( 1 ) + ¯ ∂ x ( 1 ) = ∂ v 1 v 1 ∆ v 1 1 ¯ ∆ = 0: x 0 stable x 0 = 1 ∆ = ¯ ¯ ∆ ∗ : transcritical bifurcation at x 0 ∆ > ¯ ¯ ∆ ∗ : x 0 unstable x 1 2 Loss of bistability by a kinetic perturbation Robustness analysis with kinetic perturbations, S. Waldherr 11 / 26
Inducing bifurcations by kinetic perturbations Example “network” kx 2 ˙ v 1 = x = v 1 − v 2 1 + Mx 2 X v 2 = x v 1 v 2 v 2 v ( x ) ˜ v 1 ∂ ˜ ∂ x ( 1 ) + ¯ ∂ x ( 1 ) = ∂ v 1 v 1 ∆ v 1 1 ¯ ∆ = 0: x 0 stable x 0 = 1 ∆ = ¯ ¯ ∆ ∗ : transcritical bifurcation at x 0 ∆ > ¯ ¯ ∆ ∗ : x 0 unstable x 1 2 Loss of bistability by a kinetic perturbation Robustness analysis with kinetic perturbations, S. Waldherr 11 / 26
Robustness with respect to kinetic perturbations Perturbed Jacobian ˜ A (∆) = A + S ( diag v 0 )∆( diag x 0 ) − 1 Robustness radius for kinetic perturbations Robustness measure = size of smallest perturbation such that ˜ A (∆) has an eigenvalue on the imaginary axis � ∆ � | det ( j ω I − ˜ � � ψ = inf A (∆)) = 0 Can be formulated as structured singular value ( µ ) problem with G ( j ω ) = ( diag x 0 ) − 1 ( j ω I − A ) − 1 S ( diag v 0 ) ∆ � − 1 � Result: ψ = sup µ ∆ G ( j ω ) G ( j ω ) ω Robustness problem can be solved with standard µ analysis tools. Robustness analysis with kinetic perturbations, S. Waldherr 12 / 26
Robustness radius for scalar perturbations Perturbation of a single interaction: Species ℓ on reaction i 0 . . . ¯ ∆ i ℓ ( 0 . . . 1 . . . 0 ) = e i ¯ ¯ ∆ i ℓ e T ∆ = 1 ℓ . . . 0 Perturbed Jacobian ˜ A (∆ i ℓ ) = A + Se i v 0 , i ∆ i ℓ x − 1 0 ,ℓ e T ℓ Robustness radius Im G ℓ i ( j ω ) | ∆ i ℓ | | det ( j ω I − ˜ � � ψ = A (∆ i ℓ )) = 0 � − 1 � Re = sup | G ℓ i ( j ω ) | 1 ω ∈R ( G ℓ i ) ψ ℓ x − 1 µ –problem with G ℓ i ( j ω ) = e T 0 ,ℓ ( j ω I − A ) − 1 Sv 0 , i e i Explicit formula for the robustness radius Allows to detect fragile interactions in the network Robustness analysis with kinetic perturbations, S. Waldherr 13 / 26
Robustness analysis in the vector case Perturbation of several interactions either from one species ℓ to several reactions E i v or from several species E ℓ x to one reaction i . ∆ = E i ¯ ¯ ∆ i ℓ e T ℓ Perturbed Jacobian ˜ A (∆ i ℓ ) = A + SE i diag ( E i v 0 )∆ i ℓ x − 1 0 ,ℓ e T ℓ Robustness analysis in the vector case Results depend on the norm that is chosen: 1-norm or ∞ -norm: robustness radius from a linear program 2-norm: explicit formula for robustness radius Also requires to compute a supremum over ω . Non-robust perturbation is the solution of an affine equation M ¯ ∆ = b . Robustness analysis with kinetic perturbations, S. Waldherr 14 / 26
The MAPK cascade INPUT (E1) Step response for MAPKpp MAPKpp [ µ M] 0.2 MAPKKK MAPKKK* E2 0.1 MAPKK MAPKK−PP MAPKK−P MAPKK P’ase 200 400 600 time [min] MAPK−PP MAPK MAPK−P Huang & Ferrell 1996 MAPK P’ase OUTPUT Medium scale network with 22 species and 30 (irreversible) reactions Where to perturb this network in order to get oscillations? Robustness analysis with kinetic perturbations, S. Waldherr 15 / 26
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