On Estimating Derivatives of Input Signals in Biochemistry

The online estimation of the derivative of an input signal is widespread in control theory and engineering. In the realm of chemical reaction networks (CRN), this raises however a number of specific issues on the different ways to achieve it. A CRN pattern for implementing a derivative block has already been proposed for the PID control of biochemical processes, and proved correct using Tikhonov's limit theorem. In this paper, we give a detailed mathematical analysis of that CRN, thus clarifying the computed quantity and quantifying the error done as a function of the reaction kinetic parameters. In a synthetic biology perspective, we show how this can be used to design error correcting terms to compute online functions involving derivatives with CRNs. In the systems biology perspective, we give the list of models in BioModels containing (in the sense of subgraph epimorphisms) the core derivative CRN, most of which being models of oscillators and control systems in the cell, and discuss in detail two such examples: one model of the circadian clock and one model of a bistable switch

On a model of online analog computation in the cell with absolute functional robustness: algebraic characterization, function compiler and error control

The Turing completeness of continuous Chemical Reaction Networks (CRNs) states that any computable real function can be computed by a continuous CRN on a finite set of molecular species, possibly restricted to elementary reactions, i.e. with at most two reactants and mass action law kinetics. In this paper, we introduce a more stringent notion of robust online analog computation, and Absolute Functional Robustness (AFR), for the CRNs that stabilize the concentration values of some output species to the result of one function of the input species concentrations, in a perfectly robust manner with respect to perturbations of both intermediate and output species. We prove that the set of real functions stabilized by a CRN with mass action law kinetics is precisely the set of real algebraic functions. Based on this result, we present a compiler which takes as input any algebraic function (defined by one polynomial and one point for selecting one branch of the algebraic curve defined by the polynomial) and generates an abstract CRN to stabilize it. Furthermore, we provide error bounds to estimate and control the error of an unperturbed system, under the assumption that the environment inputs are driven by k-Lipschitz functions.Comment: arXiv admin note: substantial text overlap with arXiv:2206.0962

Combining explicit negation and negation by failure via Belnap's logic

AbstractThis paper deals with logic programs containing two kinds of negation: negation as failure and explicit negation. This allows two different forms of reasoning in the presence of incomplete information. Such programs have been introduced by Gelfond and Lifschitz and called extended programs. We provide them with a logical semantics in the style of Kunen, based on Belnap's four-valued logic, and an answer sets' semantics that is shown to be equivalent to that of Gelfond and Lifschitz.The proofs rely on a translation into normal programs, and on a variant of Fitting's extension of logic programming to bilattices

Information Leakage in a Music Score

International audienc

Graphical Conditions for Rate Independence in Chemical Reaction Networks

Chemical Reaction Networks (CRNs) provide a useful abstraction of molecular interaction networks in which molecular structures as well as mass conservation principles are abstracted away to focus on the main dynamical properties of the network structure. In their interpretation by ordinary differential equations, we say that a CRN with distinguished input and output species computes a positive real function $f : R+$\rightarrow$R+$, if for any initial concentration x of the input species, the concentration of the output molecular species stabilizes at concentration f (x). The Turing-completeness of that notion of chemical analog computation has been established by proving that any computable real function can be computed by a CRN over a finite set of molecular species. Rate-independent CRNs form a restricted class of CRNs of high practical value since they enjoy a form of absolute robustness in the sense that the result is completely independent of the reaction rates and depends solely on the input concentrations. The functions computed by rate-independent CRNs have been characterized mathematically as the set of piecewise linear functions from input species. However, this does not provide a mean to decide whether a given CRN is rate-independent. In this paper, we provide graphical conditions on the Petri Net structure of a CRN which entail the rate-independence property either for all species or for some output species. We show that in the curated part of the Biomodels repository, among the 590 reaction models tested, 2 reaction graphs were found to satisfy our rate-independence conditions for all species, 94 for some output species, among which 29 for some non-trivial output species. Our graphical conditions are based on a non-standard use of the Petri net notions of place-invariants and siphons which are computed by constraint programming techniques for efficiency reasons

Compiling Elementary Mathematical Functions into Finite Chemical Reaction Networks via a Polynomialization Algorithm for ODEs

The Turing completeness result for continuous chemical reaction networks (CRN) shows that any computable function over the real numbers can be computed by a CRN over a finite set of formal molecular species using at most bimolecular reactions with mass action law kinetics. The proof uses a previous result of Turing completeness for functions defined by polynomial ordinary differential equations (PODE), the dualrail encoding of real variables by the difference of concentration between two molecular species, and a back-end quadratization transformation to restrict to elementary reactions with at most two reactants. In this paper, we present a polynomialization algorithm of quadratic time complexity to transform a system of elementary differential equations in PODE. This algorithm is used as a front-end transformation to compile any elementary mathematical function, either of time or of some input species, into a finite CRN. We illustrate the performance of our compiler on a benchmark of elementary functions relevant to CRN design problems in synthetic biology specified by mathematical functions. In particular, the abstract CRN obtained by compilation of the Hill function of order 5 is compared to the natural CRN structure of MAPK signalling networks

Machinerie cellulaire et programmation biochimique: vers une informatique de la cellule

National audienceLa biologie des systèmes est un courant de recherche multi-disciplinairequi cherche à comprendre les processus biologiques complexes en termedes interactions biochimiques à l'échelle de la cellule.Pour l'informaticien, la difficulté n'est pas tant dans le nombre d'espèces moléculaireset de leurs interactions, que dans la nature non conventionnelle du calcul biochimiquequi est concurrent, distribué, partiellement analogique,et opéré par des systèmes de réactions acquis par l'évolution.Le pari de voir les cellules comme des machines, et les systèmes de réactions biochimiquescomme des programmes, lance de nouveaux défis à l'informatique fondamentaleet ouvre de nouvelles perspectives en biologie et en médecine.Dans cet exposé nous présenterons quelques concepts clés de cette démarcheet l'illustrerons par un succès obtenu pour l'élucidation de la dynamique complexede certaines voies de signalisation cellulaire

Mixture Model-CMA-ES

internship reportWe report on our attempt to improve the CMA-ES global optimization algorithm based on two ideas: first the use of Sobol's quasi-random low discrepancy numbers instead of pseudo-random numbers, second the design of an alternative to sequential restarts to dynamically adapt the population size, using a mixture model extension of CMA-ES (MM-CMA-ES). On the standard Coco benchmark for evaluating global stochastic optimization methods, the use of Sobol numbers shows a quite uniform improvement, as was already shown by Teytaud last year. On the other hand, MM-CMA-ES does not show speed-up w.r.t. CMA-ES with IPOP restart strategy, even on objective functions with many local minima such as the Rastrigin function. The reason is the overhead in the number of evaluation of the objective functions, introduced by the MM strategy, and the very subtle effect of the adaptive step size strategy of CMA-ES to escape from the covering of several local minima by one (large) normal distribution. We conclude on some perspectives for improvement

Reactmine: a search algorithm for inferring chemical reaction networks from time series data

Inferring chemical reaction networks (CRN) from time series data is a challenge encouraged by the growing availability of quantitative temporal data at the cellular level. This motivates the design of algorithms to infer the preponderant reactions between the molecular species observed in a given biochemical process, and help to build CRN model structure and kinetics. Existing ODE-based inference methods such as SINDy resort to least square regression combined with sparsity-enforcing penalization, such as Lasso. However, when the input time series are only available in wild type conditions in which all reactions are present, we observe that current methods fail to learn sparse models. Results: We present Reactmine, a CRN learning algorithm which enforces sparsity by inferring reactions in a sequential fashion within a search tree of bounded depth, ranking the inferred reaction candidates according to the variance of their kinetics, and re-optimizing the CRN kinetic parameters on the whole trace in a final pass to rank the inferred CRN candidates. We first evaluate its performance on simulation data from a benchmark of hidden CRNs, together with algorithmic hyperparameter sensitivity analyses, and then on two sets of real experimental data: one from protein fluorescence videomicroscopy of cell cycle and circadian clock markers, and one from biomedical measurements of systemic circadian biomarkers possibly acting on clock gene expression in peripheral organs. We show that Reactmine succeeds both on simulation data by retrieving hidden CRNs where SINDy fails, and on the two real datasets by inferring reactions in agreement with previous studies