Differential Algebra and System Modeling in Cellular Biology

Boulier, François; Lemaire, François

doi:10.1007/978-3-540-85101-1_3

Cited by 8 publications

(6 citation statements)

References 29 publications

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“…The differential elimination rewrites the inputted system of differential equations to another equivalent system according to ranking (order of terms). Here, we provide an example of differential elimination, as shown below, according to Boulier [ 3 , 5 ].…”

Section: Resultsmentioning

confidence: 99%

“…Differential elimination rewrites a system of original differential equations into an equivalent system. The rewriting feature was applied to solve the parameter optimization issue, especially in network dynamics including unmeasured variables [ 3 , 5 ], and in the applications, the equations rewritten by differential elimination were utilized to estimate the initial values for the parameter optimization, by Newton-type numerical optimization.…”

Section: Introductionmentioning

confidence: 99%

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Parameter optimization by using differential elimination: a general approach for introducing constraints into objective functions

et al. 2010

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BackgroundThe investigation of network dynamics is a major issue in systems and synthetic biology. One of the essential steps in a dynamics investigation is the parameter estimation in the model that expresses biological phenomena. Indeed, various techniques for parameter optimization have been devised and implemented in both free and commercial software. While the computational time for parameter estimation has been greatly reduced, due to improvements in calculation algorithms and the advent of high performance computers, the accuracy of parameter estimation has not been addressed. ResultsWe propose a new approach for parameter optimization by using differential elimination, to estimate kinetic parameter values with a high degree of accuracy. First, we utilize differential elimination, which is an algebraic approach for rewriting a system of differential equations into another equivalent system, to derive the constraints between kinetic parameters from differential equations. Second, we estimate the kinetic parameters introducing these constraints into an objective function, in addition to the error function of the square difference between the measured and estimated data, in the standard parameter optimization method. To evaluate the ability of our method, we performed a simulation study by using the objective function with and without the newly developed constraints: the parameters in two models of linear and non-linear equations, under the assumption that only one molecule in each model can be measured, were estimated by using a genetic algorithm (GA) and particle swarm optimization (PSO). As a result, the introduction of new constraints was dramatically effective: the GA and PSO with new constraints could successfully estimate the kinetic parameters in the simulated models, with a high degree of accuracy, while the conventional GA and PSO methods without them frequently failed.ConclusionsThe introduction of new constraints in an objective function by using differential elimination resulted in the drastic improvement of the estimation accuracy in parameter optimization methods. The performance of our approach was illustrated by simulations of the parameter optimization for two models of linear and non-linear equations, which included unmeasured molecules, by two types of optimization techniques. As a result, our method is a promising development in parameter optimization.

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Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Parameter optimization by using differential elimination: a general approach for introducing constraints into objective functions

et al. 2010

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“…For doing this, one typically abstracts from functions (analytic, meromorphic, etc) to elements of differential fields (fields equipped with a derivation or several commuting derivations). This approach turned out to be fruitful yielding interesting results from theoretical and applied perspectives (see, e.g., [2,5,6,16,21]). Furthermore, one can additionally use powerful tools from model theory to study differential fields (see, e.g., [14,15]).…”

Section: Introductionmentioning

confidence: 99%

From algebra to analysis: new proofs of theorems by Ritt and Seidenberg

Павлов¹,

Pogudin²,

Yury³

2021

Preprint

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Ritt's theorem of zeroes and Seidenberg's embedding theorem are classical results in differential algebra allowing to connect algebraic and modeltheoretic results on nonlinear PDEs to the realm of analysis. However, the existing proofs of these results use sophisticated tools from constructive algebra (characteristic set theory) and analysis (Riquier's existence theorem).In this paper, we give new short proofs for both theorems relying only on basic facts from differential algebra and the classical Cauchy-Kovalevskaya theorem for PDEs.

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“…Hence, Groebner Basis Theory. Groebner Basis Theory has been used to solve problems in statistics [25], [23], biology and chemistry [2], [3], [16] geometric theorem proving [30], [5], railway systems [24], and Downloaded by [Selcuk Universitesi] at 04:36 03 January 2015 robotics [5], [22], [7], [9]. Developed by Bruno Buchberger [4], an algebraic algorithm can be applied to a given set of non-zero polynomials, producing a basis set, such that every polynomial in the original set is a linear combination of those polynomials in the basis set.…”

Section: Introductionmentioning

confidence: 99%

“…c By solving for 1 c , we fi nd two solutions c 11 and c 12 , and for each, there is a corresponding value for s 3 By using Groebner Basis Theory and MAGMA, we have found real solutions (see above) to the inverse kinematics robotics problem. In fact, we have found all of the possible formations to place the robot hand at the point ( , ) a b .…”

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confidence: 98%

A kinematic analysis of the gmf a-510 robot: An introduction and application of groebner basis theory

Kendricks

2013

Journal of Interdisciplinary Mathematics

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Differential Algebra and System Modeling in Cellular Biology

Cited by 8 publications

References 29 publications

Parameter optimization by using differential elimination: a general approach for introducing constraints into objective functions

Parameter optimization by using differential elimination: a general approach for introducing constraints into objective functions

From algebra to analysis: new proofs of theorems by Ritt and Seidenberg

A kinematic analysis of the gmf a-510 robot: An introduction and application of groebner basis theory

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