Polynomial dynamical systems in systems biology

Stigler, Brandilyn

doi:10.1090/psapm/064/2359649

Cited by 14 publications

(15 citation statements)

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“…Good references for the algebra are [7] and [21], where clear definitions of the colon ideal, prime or minimal decomposition, radical ideal, and reduced or normal form are given. Much of the algebra is also presented in [30] for related applications in statistics, and the use of algebra for dynamics in biological networks is explained in [22] and [37]. …”

Section: Resultsmentioning

confidence: 99%

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Monomials and Basin Cylinders for Network Dynamics

Austin¹,

Dinwoodie²

2015

SIAM J. Appl. Dyn. Syst.

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We describe methods to identify cylinder sets inside a basin of attraction for Boolean dynamics of biological networks. Such sets are used for designing regulatory interventions that make the system evolve towards a chosen attractor, for example initiating apoptosis in a cancer cell. We describe two algebraic methods for identifying cylinders inside a basin of attraction, one based on the Groebner fan that finds monomials that define cylinders and the other on primary decomposition. Both methods are applied to current examples of gene networks.

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Section: Resultsmentioning

confidence: 99%

“…A Boolean network is a discrete dynamical system with binary state space which evolves over time using a transition map F . Biological modeling with Boolean networks goes back at least to Thomas [39], and they continue to be widely used [2], [20], [27], [35], [34], [36], [37]. …”

Section: Introductionmentioning

confidence: 99%

Monomials and Basin Cylinders for Network Dynamics

Austin¹,

Dinwoodie²

2015

SIAM J. Appl. Dyn. Syst.

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“…The ideal-variety correspondence [7] allows one to move between the two and find geometric properties of the set of points from algebraic properties of the ideal. To use commutative algebra for dynamics, one must write the dynamics as polynomial functions (see [22] and [38] for background). For example, the dynamics in Figure 1 would be written f 1 ( s 1 , s 2 ) = s 1 + s 2 −2 s 1 · s 2 , f 2 ( s 1 , s 2 ) = s 1 where the s 1 , s 2 are indeterminates for a polynomial ring ℂ[ s 1 , s 2 ], technically not the same as state variables x 1 , x 2 which are merely symbols for unspecified values of 0 or 1.…”

Section: Algebraic Computationmentioning

confidence: 99%

Computational methods for asynchronous basins

Dinwoodie¹

2016

DCDS-B

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For a Boolean network we consider asynchronous updates and define the exclusive asynchronous basin of attraction for any steady state or cyclic attractor. An algorithm based on commutative algebra is presented to compute the exclusive basin. Finally its use for targeting desirable attractors by selective intervention on network nodes is illustrated with two examples, one cell signalling network and one sensor network measuring human mobility.

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“…, x n ) ∈ R n . Many models of f i exist, and polynomial dynamical systems (PDSs) have been extensively studied [6,18,22,[30][31][32]43,47]. In this case, f i are described by polynomials (i.e., linear combinations of monomials).…”

Section: Introductionmentioning

confidence: 99%

Noise-tolerant algebraic method for reconstruction of nonlinear dynamical systems

Kera

Hasegawa

2016

Nonlinear Dyn

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Reconstruction of nonlinear dynamical systems from noisy time series data has attracted much attention in recent years, and it has broad applications in many fields, especially in theoretical biology. Here, we propose a reconstruction algorithm based on the noise-tolerant algebraic approach that can infer underlying continuous-time polynomial dynamical systems from the observed time series data. The advantages of our approach are that it can handle nonlinearity, and it can be applied to noisy systems. Our approach reduces the range of monomials to be considered without any a priori knowledge as many other studies does. We apply the proposed method to two oscillatory systems (the FitzHugh-Nagumo and the Oregonator) subject to noise, where the conventional fitting methods based on L 1 and L 2 regularizations often fail. Our numerical experiments show that the proposed method accurately reconstructs the continuous-time polynomial dynamical systems. We also apply the method to a chaotic system and show the effectiveness of our approach.

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Polynomial dynamical systems in systems biology

Cited by 14 publications

References 0 publications

Monomials and Basin Cylinders for Network Dynamics

Monomials and Basin Cylinders for Network Dynamics

Computational methods for asynchronous basins

Noise-tolerant algebraic method for reconstruction of nonlinear dynamical systems

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