Tracer particle diffusion in a system with hardcore interacting particles

In this study, inspired by the work of K. Nakazato and K. Kitahara [Prog. Theor. Phys. 64, 2261 (1980)], we consider the theoretical problem of tracer particle diffusion in an environment of diffusing hardcore interacting crowder particles. The tracer particle has a different diffusion constant from the crowder particles. Based on a transformation of the generating function, we provide an exact formal expansion for the tracer particle probability density, valid for any lattice in the thermodynamic limit. By applying this formal solution to dynamics on regular Bravais lattices we provide a closed form approximation for the tracer particle diffusion constant which extends the Nakazato and Kitahara results to include also b.c.c. and f.c.c. lattices. Finally, we compare our analytical results to simulations in two and three dimensions.Comment: 28 pages with appendix, 5 figure. To appear in JSTA

Fitting a function to time-dependent ensemble averaged data

Time-dependent ensemble averages, i.e., trajectory-based averages of some observable, are of importance in many fields of science. A crucial objective when interpreting such data is to fit these averages (for instance, squared displacements) with a function and extract parameters (such as diffusion constants). A commonly overlooked challenge in such function fitting procedures is that fluctuations around mean values, by construction, exhibit temporal correlations. We show that the only available general purpose function fitting methods, correlated chi-square method and the weighted least squares method (which neglects correlation), fail at either robust parameter estimation or accurate error estimation. We remedy this by deriving a new closed-form error estimation formula for weighted least square fitting. The new formula uses the full covariance matrix, i.e., rigorously includes temporal correlations, but is free of the robustness issues, inherent to the correlated chi-square method. We demonstrate its accuracy in four examples of importance in many fields: Brownian motion, damped harmonic oscillation, fractional Brownian motion and continuous time random walks. We also successfully apply our method, weighted least squares including correlation in error estimation (WLS-ICE), to particle tracking data. The WLS-ICE method is applicable to arbitrary fit functions, and we provide a publically available WLS-ICE software.Comment: 47 pages (main text: 15 pages, supplementary: 32 pages

Aging dynamics in interacting many-body systems

Low-dimensional, complex systems are often characterized by logarithmically slow dynamics. We study the generic motion of a labeled particle in an ensemble of identical diffusing particles with hardcore interactions in a strongly disordered, one-dimensional environment. Each particle in this single file is trapped for a random waiting time $\tau$ with power law distribution $\psi(\tau)\simeq\tau^{-1- \alpha}$, such that the $\tau$ values are independent, local quantities for all particles. From scaling arguments and simulations, we find that for the scale-free waiting time case $0<\alpha<1$, the tracer particle dynamics is ultra-slow with a logarithmic mean square displacement (MSD) $\langle x^2(t)\rangle\simeq(\log t)^{1/2}$. This extreme slowing down compared to regular single file motion $\langle x^2(t)\rangle\simeq t^{1/2}$ is due to the high likelihood that the labeled particle keeps encountering strongly immobilized neighbors. For the case $1<\alpha<2$ we observe the MSD scaling $\langle x^2(t)\rangle\simeq t^{\gamma}$, where $\gamma2$ we recover Harris law $\simeq t^{1/2}$.Comment: 5 pages, 4 figure

On gene regulatory networks and data fitting

Living organisms can be viewed as complex biological machines. In order to function, they must regulate their internal mechanism to do the right thing, at the right time, and in the right amount. Part of this regulation is encoded in gene regulatory networks. These are built up of genes which produce special proteins (transcription factors, tf) that regulate other tf-producing genes. Thus a network is formed with genes (nodes) linked together by their mutual regulation (edges).By constructing simplified models, we investigate such gene networks. The models allow us to probe general principles behind what shapes these networks (paper II), as well as specific networks such as that which endows the plant Arabidopsis thaliana with the ability to predict dawn and dusk (paper III). We also present a model for dynamically generating transcriptional networks which encode function from a single variable-length binary representation of dna (string of ones and zeroes). This gives a natural way for the network to evolve by mutations. However, performing a meaningful and efficient crossover operation on two dna strings of different length becomes a challenge. We address this by introducing a heuristic algorithm, which we compare against existing methods (paper IV).Additionally, we present a correct error estimation for the popular least squares method that is valid also for nonlinear functions applied to highly correlated data (paper I). For model fitting to correlated data, one has previously been constrained to use either a maximum likelihood approach, which leads to strong bias in the estimated parameters, or a least squares approach, which gives an incorrect error estimate. We also derive the first order contribution of the bias for both the maximum likelihood and the least squares method, and introduce a minimum variance function fitting method suited for Brownian motion

Tracer Particle Dynamics in Heterogeneous Many-body Systems

By use of a lattice random walk algorithm we model diffusion in a many-body system and study the mean square displacement (MSD) for a tagged particle for different distributions of crowding particles, with particular emphasis on obtaining the correlation factor which contains the corrections to the mean-field result in such a system. The MSD in such a crowded environment is investigated and we find that the analytical correlation factor developed by Nakazato et al.1 is not accurate for a tracer particle that is faster than the surrounding homogeneous crowding particles. Simulation results for the correlation factor is found for diffusion in a heterogeneous environment, where the friction coefficients of the crowding particles were drawn from a uniform distribution, and a power-law distribution. The simulation results can not be fitted to Nakazato’s analytical form for the correlation factor. The MSD of a particle with the same diffusion constant as the crowding particles is investigated for a system where the particles have a probability, proportional to the corresponding Boltzmann factor, to form bonds to their nearest-neighbors. The MSD is found to be subdiffusive, and the exponent decreases almost linearly with increasing interaction strength and is roughly independent on the concentration of crowding particles. Department of Astronomy and Theoretical Physics Lun

Rethinking transcriptional activation in the Arabidopsis circadian clock.

Circadian clocks are biological timekeepers that allow living cells to time their activity in anticipation of predictable daily changes in light and other environmental factors. The complexity of the circadian clock in higher plants makes it difficult to understand the role of individual genes or molecular interactions, and mathematical modelling has been useful in guiding clock research in model organisms such as Arabidopsis thaliana. We present a model of the circadian clock in Arabidopsis, based on a large corpus of published time course data. It appears from experimental evidence in the literature that most interactions in the clock are repressive. Hence, we remove all transcriptional activation found in previous models of this system, and instead extend the system by including two new components, the morning-expressed activator RVE8 and the nightly repressor/activator NOX. Our modelling results demonstrate that the clock does not need a large number of activators in order to reproduce the observed gene expression patterns. For example, the sequential expression of the PRR genes does not require the genes to be connected as a series of activators. In the presented model, transcriptional activation is exclusively the task of RVE8. Predictions of how strongly RVE8 affects its targets are found to agree with earlier interpretations of the experimental data, but generally we find that the many negative feedbacks in the system should discourage intuitive interpretations of mutant phenotypes. The dynamics of the clock are difficult to predict without mathematical modelling, and the clock is better viewed as a tangled web than as a series of loops

The number of parameters and variables in different <i>Arabidopsis</i> clock models.

<p>Parameter counts in parentheses refer to constant integer Hill coefficients, which are written explicitly into the F2014 equations. Variables in parentheses for P2012 refer to ABA related variables.</p

Selection Shapes Transcriptional Logic and Regulatory Specialization in Genetic Networks.

Living organisms need to regulate their gene expression in response to environmental signals and internal cues. This is a computational task where genes act as logic gates that connect to form transcriptional networks, which are shaped at all scales by evolution. Large-scale mutations such as gene duplications and deletions add and remove network components, whereas smaller mutations alter the connections between them. Selection determines what mutations are accepted, but its importance for shaping the resulting networks has been debated

The effects of RVE8 in the model.

<p>(A–C) Expression levels in the transition from LD to LL, comparing the model (eight parameter sets, solid black lines) with experimental data (green triangles <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003705#pcbi.1003705-Hsu1" target="_blank">[29]</a>, red squares <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003705#pcbi.1003705-GongWHe1" target="_blank">[59]</a>, blue circles <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003705#pcbi.1003705-Rawat1" target="_blank">[28]</a> and purple diamonds <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003705#pcbi.1003705-Hsu2" target="_blank">[60]</a>). (A) <i>RVE8</i> mRNA in wt, (B) <i>LHY</i> in wt, and (C) <i>LHY</i> in <i>rve4</i>;<i>rve6</i>;<i>rve8</i>. (D–F) The effect of RVE8 on each of its target genes, as a time-dependent multiplicative factor, in the eight parameter sets. (D) <i>PRR9</i> (solid red) and <i>PRR5</i> (dotted blue), (E) <i>GI</i> (solid green) and <i>TOC1</i> (dotted black), and (F) <i>LUX</i> (solid purple) and <i>ELF4</i> (dotted light blue).</p

Expression and regulation of the PRR genes.

<p>(A–C) The mRNA levels of <i>PRR9</i> (solid red), <i>PRR7</i> (long dashed green), <i>PRR5</i> (short dashed blue) and <i>TOC1</i> (dotted black) in the transition from LD to LL. (A) The F2014 model with eight different parameter sets. (B) Experimental data: <i>PRR9 </i><a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003705#pcbi.1003705-Nakamichi2" target="_blank">[35]</a>, <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003705#pcbi.1003705-Ding1" target="_blank">[36]</a>, <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003705#pcbi.1003705-Dixon1" target="_blank">[54]</a>, <i>PRR7 </i><a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003705#pcbi.1003705-Nakamichi2" target="_blank">[35]</a>, <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003705#pcbi.1003705-Dixon1" target="_blank">[54]</a>, <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003705#pcbi.1003705-Baudry1" target="_blank">[55]</a>, <i>PRR5 </i><a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003705#pcbi.1003705-Hsu1" target="_blank">[29]</a>, <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003705#pcbi.1003705-Baudry1" target="_blank">[55]</a>, <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003705#pcbi.1003705-Matsushika2" target="_blank">[56]</a> and <i>TOC1 </i><a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003705#pcbi.1003705-Edwards1" target="_blank">[53]</a>, <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003705#pcbi.1003705-Farr2" target="_blank">[57]</a>, <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003705#pcbi.1003705-Sato1" target="_blank">[58]</a>. (C) The P2012 and P2011 models (thick and thin lines, respectively). (D) Total PRR5 protein level in <i>prr9;prr7</i> in LD in F2014 (solid black), P2011 (dashed red), P2012 (dotted blue) and experimental data (green triangles <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003705#pcbi.1003705-Dixon1" target="_blank">[54]</a>). (E) The predicted repression of <i>PRR</i> transcription by CCA1 and LHY, as a multiplicative factor, with colours as in (A–C). (F) <i>PRR9</i> mRNA in <i>cca1-11</i>;<i>lhy-21</i> in LD, normalized to the corresponding wt curves in (A–C); colours as in (D) but data from <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003705#pcbi.1003705-Nakamichi1" target="_blank">[11]</a>. The peak levels in (A), (C) and (D) were normalized to 1, whereas the levels in (B) were adjusted manually.</p