39 research outputs found
Mathematical explanation of the predictive power of the X-level approach reaction noise estimator method
The X-level Approach Reaction Noise Estimator (XARNES) method has been developed previously to study reaction noise in well mixed reaction volumes. The method is a typical moment closure method and it works by closing the infinite hierarchy of equations that describe moments of the particle number distribution function. This is done by using correlation forms which describe correlation effects in a strict mathematical way. The variable X is used to specify which correlation effects (forms) are included in the description. Previously, it was argued, in a rather informal way, that the method should work well in situations where the particle number distribution function is Poisson-like. Numerical tests confirmed this. It was shown that the predictive power of the method increases, i.e. the agreement between the theory and simulations improves, if X is increased. In here, these features of the method are explained by using rigorous mathematical reasoning. Three derivative matching theoremsare proven which show that the observed numerical behavior is generic to the method
Safe uses of Hill's model: an exact comparison with the Adair-Klotz model
<p>Abstract</p> <p>Background</p> <p>The Hill function and the related Hill model are used frequently to study processes in the living cell. There are very few studies investigating the situations in which the model can be safely used. For example, it has been shown, at the mean field level, that the dose response curve obtained from a Hill model agrees well with the dose response curves obtained from a more complicated Adair-Klotz model, provided that the parameters of the Adair-Klotz model describe strongly cooperative binding. However, it has not been established whether such findings can be extended to other properties and non-mean field (stochastic) versions of the same, or other, models.</p> <p>Results</p> <p>In this work a rather generic quantitative framework for approaching such a problem is suggested. The main idea is to focus on comparing the particle number distribution functions for Hill's and Adair-Klotz's models instead of investigating a particular property (e.g. the dose response curve). The approach is valid for any model that can be mathematically related to the Hill model. The Adair-Klotz model is used to illustrate the technique. One main and two auxiliary similarity measures were introduced to compare the distributions in a quantitative way. Both time dependent and the equilibrium properties of the similarity measures were studied.</p> <p>Conclusions</p> <p>A strongly cooperative Adair-Klotz model can be replaced by a suitable Hill model in such a way that any property computed from the two models, even the one describing stochastic features, is approximately the same. The quantitative analysis showed that boundaries of the regions in the parameter space where the models behave in the same way exhibit a rather rich structure.</p
Two-Species Reaction-Diffusion System with Equal Diffusion Constants: Anomalous Density Decay at Large Times
We study a two-species reaction-diffusion model where A+A->0, A+B->0 and
B+B->0, with annihilation rates lambda0, delta0 > lambda0 and lambda0,
respectively. The initial particle configuration is taken to be randomly mixed
with mean densities nA(0) > nB(0), and with the two species A and B diffusing
with the same diffusion constant. A field-theoretic renormalization group
analysis suggests that, contrary to expectation, the large-time density of the
minority species decays at the same rate as the majority when d<=2. Monte Carlo
data supports the field theory prediction in d=1, while in d=2 the
logarithmically slow convergence to the large-time asymptotics makes a
numerical test difficult.Comment: revised version (more figures, claim on exactnes of d=2 treatment
removed), 5 pages, 3 figures, RevTex, see related paper Phys. Rev. E, R3787,
(1999) or cond-mat/9901147, to appear in Phys. Rev.
A danger of low copy numbers for inferring incorrect cooperativity degree
Background: A dose-response curve depicts fraction of bound proteins as a function of unbound ligands. Dose-response curves are used to measure the cooperativity degree of a ligand binding process. Frequently, the Hill function is used to fit the experimental data. The Hill function is parameterized by the value of the dissociation constant, and the Hill coefficient which describes the cooperativity degree. The use of Hill's model and the Hill function have been heavily criticised in this context, predominantly the assumption that all ligands bind at once, which lead to further refinements of the model. In this work, the validity of the Hill function has been studied from an entirely different point of view. In the limit of low copy numbers the dynamics of the system becomes noisy. The goal was to asses the validity of the Hill function in this limit, and to see in which ways the effects of the fluctuations change the form of the dose-response curves.
Results: Dose-response curves were computed taking into account effects of fluctuations. The effects of fluctuations were described at the lowest order (the second moment of the particle number distribution) by using previously developed Pair Approach Reaction Noise EStimator (PARNES) method. The stationary state of the system is described by nine equations with nine unknowns. To obtain fluctuation corrected dose-response curves the equations have been investigated numerically.
Conclusions: The Hill function cannot describe dose-response curves in a low particle limit. First, dose-response curves are not solely parameterized by the dissociation constant and the Hill coefficient. In general, the shape of a dose-response curve depends on the variables that describe how an experiment (ensemble) is designed. Second, dose-response curves are multi valued in a rather non-trivial way
Diffusive transport in networks built of containers and tubes
We developed analytical and numerical methods to study a transport of
non-interacting particles in large networks consisting of M d-dimensional
containers C_1,...,C_M with radii R_i linked together by tubes of length l_{ij}
and radii a_{ij} where i,j=1,2,...,M. Tubes may join directly with each other
forming junctions. It is possible that some links are absent. Instead of
solving the diffusion equation for the full problem we formulated an approach
that is computationally more efficient. We derived a set of rate equations that
govern the time dependence of the number of particles in each container
N_1(t),N_2(t),...,N_M(t). In such a way the complicated transport problem is
reduced to a set of M first order integro-differential equations in time, which
can be solved efficiently by the algorithm presented here. The workings of the
method have been demonstrated on a couple of examples: networks involving
three, four and seven containers, and one network with a three-point junction.
Already simple networks with relatively few containers exhibit interesting
transport behavior. For example, we showed that it is possible to adjust the
geometry of the networks so that the particle concentration varies in time in a
wave-like manner. Such behavior deviates from simple exponential growth and
decay occurring in the two container system.Comment: 21 pages, 18 figures, REVTEX4; new figure added, reduced emphasis on
graph theory, additional discussion added (computational cost, one
dimensional tubes
Ethics of Artificial Intelligence Demarcations
In this paper we present a set of key demarcations, particularly important
when discussing ethical and societal issues of current AI research and
applications. Properly distinguishing issues and concerns related to Artificial
General Intelligence and weak AI, between symbolic and connectionist AI, AI
methods, data and applications are prerequisites for an informed debate. Such
demarcations would not only facilitate much-needed discussions on ethics on
current AI technologies and research. In addition sufficiently establishing
such demarcations would also enhance knowledge-sharing and support rigor in
interdisciplinary research between technical and social sciences.Comment: Proceedings of the Norwegian AI Symposium 2019 (NAIS 2019),
Trondheim, Norwa
Embedding a Native State into a Random Heteropolymer Model: The Dynamic Approach
We study a random heteropolymer model with Langevin dynamics, in the
supersymmetric formulation. Employing a procedure similar to one that has been
used in static calculations, we construct an ensemble in which the affinity of
the system for a native state is controlled by a "selection temperature" T0. In
the limit of high T0, the model reduces to a random heteropolymer, while for
T0-->0 the system is forced into the native state. Within the Gaussian
variational approach that we employed previously for the random heteropolymer,
we explore the phases of the system for large and small T0. For large T0, the
system exhibits a (dynamical) spin glass phase, like that found for the random
heteropolymer, below a temperature Tg. For small T0, we find an ordered phase,
characterized by a nonzero overlap with the native state, below a temperature
Tn \propto 1/T0 > Tg. However, the random-globule phase remains locally stable
below Tn, down to the dynamical glass transition at Tg. Thus, in this model,
folding is rapid for temperatures between Tg and Tn, but below Tg the system
can get trapped in conformations uncorrelated with the native state. At a lower
temperature, the ordered phase can also undergo a dynamical glass transition,
splitting into substates separated by large barriers.Comment: 19 pages, revtex, 6 figure
Applications of Field-Theoretic Renormalization Group Methods to Reaction-Diffusion Problems
We review the application of field-theoretic renormalization group (RG)
methods to the study of fluctuations in reaction-diffusion problems. We first
investigate the physical origin of universality in these systems, before
comparing RG methods to other available analytic techniques, including exact
solutions and Smoluchowski-type approximations. Starting from the microscopic
reaction-diffusion master equation, we then pedagogically detail the mapping to
a field theory for the single-species reaction k A -> l A (l < k). We employ
this particularly simple but non-trivial system to introduce the
field-theoretic RG tools, including the diagrammatic perturbation expansion,
renormalization, and Callan-Symanzik RG flow equation. We demonstrate how these
techniques permit the calculation of universal quantities such as density decay
exponents and amplitudes via perturbative eps = d_c - d expansions with respect
to the upper critical dimension d_c. With these basics established, we then
provide an overview of more sophisticated applications to multiple species
reactions, disorder effects, L'evy flights, persistence problems, and the
influence of spatial boundaries. We also analyze field-theoretic approaches to
nonequilibrium phase transitions separating active from absorbing states. We
focus particularly on the generic directed percolation universality class, as
well as on the most prominent exception to this class: even-offspring branching
and annihilating random walks. Finally, we summarize the state of the field and
present our perspective on outstanding problems for the future.Comment: 10 figures include