431 research outputs found
Stochastic kinetic models: Dynamic independence, modularity and graphs
The dynamic properties and independence structure of stochastic kinetic
models (SKMs) are analyzed. An SKM is a highly multivariate jump process used
to model chemical reaction networks, particularly those in biochemical and
cellular systems. We identify SKM subprocesses with the corresponding counting
processes and propose a directed, cyclic graph (the kinetic independence graph
or KIG) that encodes the local independence structure of their conditional
intensities. Given a partition of the vertices, the graphical
separation in the undirected KIG has an intuitive chemical
interpretation and implies that is locally independent of given . It is proved that this separation also results in global independence of
the internal histories of and conditional on a history of the jumps in
which, under conditions we derive, corresponds to the internal history of
. The results enable mathematical definition of a modularization of an SKM
using its implied dynamics. Graphical decomposition methods are developed for
the identification and efficient computation of nested modularizations.
Application to an SKM of the red blood cell advances understanding of this
biochemical system.Comment: Published in at http://dx.doi.org/10.1214/09-AOS779 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Composable computation in discrete chemical reaction networks
We study the composability of discrete chemical reaction networks (CRNs) that
stably compute (i.e., with probability 0 of error) integer-valued functions
. We consider output-oblivious CRNs in which the
output species is never a reactant (input) to any reaction. The class of
output-oblivious CRNs is fundamental, appearing in earlier studies of CRN
computation, because it is precisely the class of CRNs that can be composed by
simply renaming the output of the upstream CRN to match the input of the
downstream CRN.
Our main theorem precisely characterizes the functions stably computable
by output-oblivious CRNs with an initial leader. The key necessary condition is
that for sufficiently large inputs, is the minimum of a finite number of
nondecreasing quilt-affine functions. (An affine function is linear with a
constant offset; a quilt-affine function is linear with a periodic offset)
Composable Computation in Leaderless, Discrete Chemical Reaction Networks
We classify the functions f:?^d ? ? that are stably computable by leaderless, output-oblivious discrete (stochastic) Chemical Reaction Networks (CRNs). CRNs that compute such functions are systems of reactions over species that include d designated input species, whose initial counts represent an input x ? ?^d, and one output species whose eventual count represents f(x). Chen et al. showed that the class of functions computable by CRNs is precisely the semilinear functions. In output-oblivious CRNs, the output species is never a reactant. Output-oblivious CRNs are easily composable since a downstream CRN can consume the output of an upstream CRN without affecting its correctness. Severson et al. showed that output-oblivious CRNs compute exactly the subclass of semilinear functions that are eventually the minimum of quilt-affine functions, i.e., affine functions with different intercepts in each of finitely many congruence classes. They call such functions the output-oblivious functions. A leaderless CRN can compute only superadditive functions, and so a leaderless output-oblivious CRN can compute only superadditive, output-oblivious functions. In this work we show that a function f:?^d ? ? is stably computable by a leaderless, output-oblivious CRN if and only if it is superadditive and output-oblivious
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
CRNs Exposed: A Method for the Systematic Exploration of Chemical Reaction Networks
Formal methods have enabled breakthroughs in many fields, such as in hardware verification, machine learning and biological systems. The key object of interest in systems biology, synthetic biology, and molecular programming is chemical reaction networks (CRNs) which formalizes coupled chemical reactions in a well-mixed solution. CRNs are pivotal for our understanding of biological regulatory and metabolic networks, as well as for programming engineered molecular behavior. Although it is clear that small CRNs are capable of complex dynamics and computational behavior, it remains difficult to explore the space of CRNs in search for desired functionality. We use Alloy, a tool for expressing structural constraints and behavior in software systems, to enumerate CRNs with declaratively specified properties. We show how this framework can enumerate CRNs with a variety of structural constraints including biologically motivated catalytic networks and metabolic networks, and seesaw networks motivated by DNA nanotechnology. We also use the framework to explore analog function computation in rate-independent CRNs. By computing the desired output value with stoichiometry rather than with reaction rates (in the sense that X ? Y+Y computes multiplication by 2), such CRNs are completely robust to the choice of reaction rates or rate law. We find the smallest CRNs computing the max, minmax, abs and ReLU (rectified linear unit) functions in a natural subclass of rate-independent CRNs where rate-independence follows from structural network properties
Rate-Independent Computation in Continuous Chemical Reaction Networks
Coupled chemical interactions in a well-mixed solution are commonly
formalized as chemical reaction networks (CRNs). However, despite the
widespread use of CRNs in the natural sciences, the range of computational
behaviors exhibited by CRNs is not well understood. Here we study the following
problem: what functions f:R^k --> R can be computed by a CRN, in which the CRN
eventually produces the correct amount of the "output" molecule, no matter the
rate at which reactions proceed? This captures a previously unexplored, but
very natural class of computations: for example, the reaction X1 + X2 --> Y can
be thought to compute the function y = min(x1, x2). Such a CRN is robust in the
sense that it is correct no matter the kinetic model of chemistry, so long as
it respects the stoichiometric constraints.
We develop a reachability relation based on "what could happen" if reaction
rates can vary arbitrarily over time. We define *stable computation*
analogously to probability 1 computation in distributed computing, and connect
it with a seemingly stronger notion of rate-independent computation based on
convergence under a wide class of generalized rate laws. We also consider the
"dual-rail representation" that can represent negative values as the difference
of two concentrations and allows the composition of CRN modules. We prove that
a function is rate-independently computable if and only if it is piecewise
linear (with rational coefficients) and continuous (dual-rail representation),
or non-negative with discontinuities occurring only when some inputs switch
from zero to positive (direct representation). The many contexts where
continuous piecewise linear functions are powerful targets for implementation,
combined with the systematic construction we develop for computing these
functions, demonstrate the potential of rate-independent chemical computation.Comment: preliminary version appeared in ITCS 2014:
http://doi.org/10.1145/2554797.255482
Experimental Biological Protocols with Formal Semantics
Both experimental and computational biology is becoming increasingly
automated. Laboratory experiments are now performed automatically on
high-throughput machinery, while computational models are synthesized or
inferred automatically from data. However, integration between automated tasks
in the process of biological discovery is still lacking, largely due to
incompatible or missing formal representations. While theories are expressed
formally as computational models, existing languages for encoding and
automating experimental protocols often lack formal semantics. This makes it
challenging to extract novel understanding by identifying when theory and
experimental evidence disagree due to errors in the models or the protocols
used to validate them. To address this, we formalize the syntax of a core
protocol language, which provides a unified description for the models of
biochemical systems being experimented on, together with the discrete events
representing the liquid-handling steps of biological protocols. We present both
a deterministic and a stochastic semantics to this language, both defined in
terms of hybrid processes. In particular, the stochastic semantics captures
uncertainties in equipment tolerances, making it a suitable tool for both
experimental and computational biologists. We illustrate how the proposed
protocol language can be used for automated verification and synthesis of
laboratory experiments on case studies from the fields of chemistry and
molecular programming
- …