16 research outputs found
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
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 \rightarrow, 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
On the Complexity of Quadratization for Polynomial Differential Equations
Chemical reaction networks (CRNs) are a standard formalism used in chemistry
and biology to reason about the dynamics of molecular interaction networks. In
their interpretation by ordinary differential equations, CRNs provide a
Turing-complete model of analog computattion, in the sense that any computable
function over the reals can be computed by a finite number of molecular species
with a continuous CRN which approximates the result of that function in one of
its components in arbitrary precision. The proof of that result is based on a
previous result of Bournez et al. on the Turing-completeness of polyno-mial
ordinary differential equations with polynomial initial conditions (PIVP). It
uses an encoding of real variables by two non-negative variables for
concentrations, and a transformation to an equivalent quadratic PIVP (i.e. with
degrees at most 2) for restricting ourselves to at most bimolecular reactions.
In this paper, we study the theoretical and practical complexities of the
quadratic transformation. We show that both problems of minimizing either the
number of variables (i.e., molecular species) or the number of monomials (i.e.
elementary reactions) in a quadratic transformation of a PIVP are NP-hard. We
present an encoding of those problems in MAX-SAT and show the practical
complexity of this algorithm on a benchmark of quadratization problems inspired
from CRN design problems
On Chemical Reaction Network Design by a Nested Evolution Algorithm
International audienceOne goal of synthetic biology is to implement useful functions with biochemical reactions, either by reprogramming living cells or programming artificial vesicles. In this perspective, we consider Chemical Reaction Networks (CRN) as a programming language, and investigate the CRN program synthesis problem. Recent work has shown that CRN interpreted by differential equations are Turing-complete and can be seen as analog computers where the molecular concentrations play the role of information carriers. Any real function that is computable by a Turing machine in arbitrary precision can thus be computed by a CRN over a finite set of molecular species. The proof of this result gives a numerical method to generate a finite CRN for implementing a real function presented as the solution of a Polynomial Initial Values Problem (PIVP). In this paper, we study an alternative method based on artificial evolution to build a CRN that approximates a real function given on finite sets of input values. We present a nested search algorithm that evolves the structure of the CRN and optimizes the kinetic parameters at each generation. We evaluate this algorithm on the Heaviside and Cosine functions both as functions of time and functions of input molecular species. We then compare the CRN obtained by artificial evolution both to the CRN generated by the numerical method from a PIVP definition of the function, and to the natural CRN found in the BioModels repository for switches and oscillators
Graphical Conditions for Rate Independence in Chemical Reaction Networks
International audienceChemical 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 , 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