1,091 research outputs found
Fixed points of Boolean networks, guessing graphs, and coding theory
n this paper, we are interested in the number of fixed points of functions over a finite alphabet defined on a given signed digraph . We first use techniques from network coding to derive some lower bounds on the number of fixed points that only depends on . We then discover relationships between the number of fixed points of and problems in coding theory, especially the design of codes for the asymmetric channel. Using these relationships, we derive upper and lower bounds on the number of fixed points, which significantly improve those given in the literature. We also unveil some interesting behavior of the number of fixed points of functions with a given signed digraph when the alphabet varies. We finally prove that signed digraphs with more (disjoint) positive cycles actually do not necessarily have functions with more fixed points
Logic Integer Programming Models for Signaling Networks
We propose a static and a dynamic approach to model biological signaling
networks, and show how each can be used to answer relevant biological
questions. For this we use the two different mathematical tools of
Propositional Logic and Integer Programming. The power of discrete mathematics
for handling qualitative as well as quantitative data has so far not been
exploited in Molecular Biology, which is mostly driven by experimental
research, relying on first-order or statistical models. The arising logic
statements and integer programs are analyzed and can be solved with standard
software. For a restricted class of problems the logic models reduce to a
polynomial-time solvable satisfiability algorithm. Additionally, a more dynamic
model enables enumeration of possible time resolutions in poly-logarithmic
time. Computational experiments are included
Asynchronous simulation of Boolean networks by monotone Boolean networks
International audienceWe prove that the fully asynchronous dynamics of a Boolean network f : {0, 1}^n → {0, 1}^n without negative loop can be simulated, in a very specific way, by a monotone Boolean network with 2n components. We then use this result to prove that, for every even n, there exists a monotone Boolean network f : {0, 1}^n → {0, 1}^n , an initial configuration x and a fixed point y of f such that: (i) y can be reached from x with a fully asynchronous updating strategy, and (ii) all such strategies contains at least 2^{n/2} updates. This contrasts with the following known property: if f : {0, 1}^n → {0, 1}^n is monotone, then, for every initial configuration x, there exists a fixed point y such that y can be reached from x with a fully asynchronous strategy that contains at most n updates
Complexity of fixed point counting problems in Boolean Networks
A Boolean network (BN) with components is a discrete dynamical system
described by the successive iterations of a function . This model finds applications in biology, where fixed points play a
central role. For example, in genetic regulations, they correspond to cell
phenotypes. In this context, experiments reveal the existence of positive or
negative influences among components: component has a positive (resp.
negative) influence on component meaning that tends to mimic (resp.
negate) . The digraph of influences is called signed interaction digraph
(SID), and one SID may correspond to a large number of BNs (which is, in
average, doubly exponential according to ). The present work opens a new
perspective on the well-established study of fixed points in BNs. When
biologists discover the SID of a BN they do not know, they may ask: given that
SID, can it correspond to a BN having at least/at most fixed points?
Depending on the input, we prove that these problems are in or
complete for , ,
\textrm{NP}^{\textrm{#P}} or . In particular, we prove
that it is -complete (resp. -complete) to
decide if a given SID can correspond to a BN having at least two fixed points
(resp. no fixed point).Comment: 43 page
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