Numerical and Experimental Analysis of the p53-mdm2 Regulatory Pathway

Leeuwen, Ingeborg M.M. van; Sanders, Ian L.; Staples, Oliver; Laı́n, Sonia; Munro, Alastair J.

doi:10.1007/978-3-642-14859-0_20

Cited by 6 publications

(7 citation statements)

References 45 publications

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“…Another model we are interested in is for the damped oscillation of the p53-mdm2 regulatory pathway which is given by (see [20])

\begin{matrix} {\dot{P}}_{I} = s_{p} + j_{a} P_{A} - (d_{p} + k_{a} S (t)) P_{I} - k_{c} P_{I} M + j_{c} C, \\ \dot{M} = s_{m 0} + \frac{s_{m 1} P_{I} + s_{m 2} P_{A}}{P_{I} + P_{A} + K_{m}} + k_{u} C + j_{c} C - (d_{m} + k_{c} P_{I}) M, \\ \dot{C} = k_{c} P_{I} M - (j_{c} + k_{u}) C, \\ {\dot{P}}_{A} = k_{a} S (t) P_{I} - (j_{a} + d_{p}) P_{A}, \end{matrix}

where P I represents the concentration of the p53 tumour suppressor, M (mdm2) is the concentration of the p53's main negative regulator, C is the concentration of the p53-mdm2 complex, P A is the concentration of an active form of p53 that is resistant against mdm2-mediated degradation, S ( t ) is a transient stress stimulus which has the form S ( t ) = − e c s t , c s = γk u , s ∗ (∗ = p , m 0, m 1) are de novo synthesis rates, k ∗ (∗ = a , c , u ) are production rates, j ∗ (∗ = a , c ) are reverse reactions (e.g., dephosphorylation), d p is the degradation rate of active p53, and K m is the saturation coefficient.…”

Section: Methodsmentioning

confidence: 99%

Splitting Strategy for Simulating Genetic Regulatory Networks

You

Liu

Musa

2014

Computational and Mathematical Methods in Medicine

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The splitting approach is developed for the numerical simulation of genetic regulatory networks with a stable steady-state structure. The numerical results of the simulation of a one-gene network, a two-gene network, and a p53-mdm2 network show that the new splitting methods constructed in this paper are remarkably more effective and more suitable for long-term computation with large steps than the traditional general-purpose Runge-Kutta methods. The new methods have no restriction on the choice of stepsize due to their infinitely large stability regions.

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“…Another model we are interested in is for the damped oscillation of the p53-mdm2 regulatory pathway which is given by (see [20])

\begin{matrix} {\dot{P}}_{I} = s_{p} + j_{a} P_{A} - (d_{p} + k_{a} S (t)) P_{I} - k_{c} P_{I} M + j_{c} C, \\ \dot{M} = s_{m 0} + \frac{s_{m 1} P_{I} + s_{m 2} P_{A}}{P_{I} + P_{A} + K_{m}} + k_{u} C + j_{c} C - (d_{m} + k_{c} P_{I}) M, \\ \dot{C} = k_{c} P_{I} M - (j_{c} + k_{u}) C, \\ {\dot{P}}_{A} = k_{a} S (t) P_{I} - (j_{a} + d_{p}) P_{A}, \end{matrix}

Section: Methodsmentioning

confidence: 99%

Splitting Strategy for Simulating Genetic Regulatory Networks

You

Liu

Musa

2014

Computational and Mathematical Methods in Medicine

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“…We adopt this model since its solutions also have a stable steady-state structure of interest. The system, given by van Leeuwen et al [ 33 ] with the small transient stress stimulus S ( t ) = 0, has the form

where P I represents the concentration of the p53 tumor suppressor, M (mdm2) is the concentration of the p53's main negative regulator, C is the concentration of the p53-mdm2 complex, P A is the concentration of an active form of p53 that is resistant against mdm2-mediated degradation, s ∗ ( ∗ = p , m 0, m 1) are de novo synthesis rates, k ∗ ( ∗ = a , c , u ) are production rates, j ∗ ( ∗ = a , c ) are reverse reactions (e.g., dephosphorylation), d p is the degradation rate of active p53, and K m is the saturation coefficient.…”

Section: Numerical Illustrationsmentioning

confidence: 99%

“…The steady state y ∗ = ( P I ∗ , M ∗ , C ∗ , P A ∗ ) is determined by the following equations:

Putting in the form ( 7 ), system ( 27 ) has the rate matrix and the function

where y = ( y 1 , y 2 , y 3 , y 4 ) T = ( P I − P I ∗ , M − M ∗ , C − C ∗ , P A − P A ∗ ) T , and

We use the parameter values (see [ 33 ]) as follows:

For simplicity, we take the small function S ( t ) ≡ 0. The system has a unique steady state ( P I ∗ , M ∗ , C ∗ , P A ∗ ) = (9.42094,0.0372868,3.49529,0).…”

Section: Numerical Illustrationsmentioning

confidence: 99%

Steady-State-Preserving Simulation of Genetic Regulatory Systems

Zhang

Ehigie

Hou

et al. 2017

Computational and Mathematical Methods in Medicine

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A novel family of exponential Runge-Kutta (expRK) methods are designed incorporating the stable steady-state structure of genetic regulatory systems. A natural and convenient approach to constructing new expRK methods on the base of traditional RK methods is provided. In the numerical integration of the one-gene, two-gene, and p53-mdm2 regulatory systems, the new expRK methods are shown to be more accurate than their prototype RK methods. Moreover, for nonstiff genetic regulatory systems, the expRK methods are more efficient than some traditional exponential RK integrators in the scientific literature.

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“…We examine natural subsystems and algebraic structure in a model of the p53-mdm2 system [35,36]. Figure 7 shows such a model of the p53-mdm2 regulatory pathway, which is important in the cellular response to ionizing radiation and can trigger self-repair or, in extreme cases, the onset of programmed cell death (apoptosis).…”

Section: (B) Krebs Cyclementioning

confidence: 99%

“…In automata models, when non-trivial sequences of inputs (or events) permute some subset Y of states, the set of sequences permuting Y will be infinite, 36 and each such sequence will determine a member of permutation group (Y)per. The pair (Y, per(Y)) can be considered a 'natural subsystem', 'reaction chain', 'pool of feedback' or 'pool of local reversibility' ( §2b).…”

Section: Permutation Groups In Recurring Transient Structures In Dynamentioning

confidence: 99%

Symmetry structure in discrete models of biochemical systems: natural subsystems and the weak control hierarchy in a new model of computation driven by interactions

Nehaniv

Rhodes

Egri-Nagy

et al. 2015

Phil. Trans. R. Soc. A.

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Interaction computing is inspired by the observation that cell metabolic/regulatory systems construct order dynamically, through constrained interactions between their components and based on a wide range of possible inputs and environmental conditions. The goals of this work are to (i) identify and understand mathematically the natural subsystems and hierarchical relations in natural systems enabling this and (ii) use the resulting insights to define a new model of computation based on interactions that is useful for both biology and computation. The dynamical characteristics of the cellular pathways studied in systems biology relate, mathematically, to the computational characteristics of automata derived from them, and their internal symmetry structures to computational power. Finite discrete automata models of biological systems such as the lac operon, the Krebs cycle and p53-mdm2 genetic regulation constructed from systems biology models have canonically associated algebraic structures (their transformation semigroups). These contain permutation groups (local substructures exhibiting symmetry) that correspond to 'pools of reversibility'. These natural subsystems are related to one another in a hierarchical manner by the notion of 'weak control'. We present natural subsystems arising from several biological examples and their weak control hierarchies in detail. Finite simple non-Abelian groups are found in biological examples and can be harnessed to realize finitary universal computation. This allows ensembles of cells to achieve any desired finitary computational transformation, depending on external inputs, via suitably constrained interactions. Based on this, interaction machines that grow and change their structure recursively are introduced and applied, providing a natural model of computation driven by interactions.In the 21st century, algebra will be as important for biology as chemistry was in the 20th century.-Günter P. Wagner

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Numerical and Experimental Analysis of the p53-mdm2 Regulatory Pathway

Cited by 6 publications

References 45 publications

Splitting Strategy for Simulating Genetic Regulatory Networks

Splitting Strategy for Simulating Genetic Regulatory Networks

Steady-State-Preserving Simulation of Genetic Regulatory Systems

Symmetry structure in discrete models of biochemical systems: natural subsystems and the weak control hierarchy in a new model of computation driven by interactions

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