arXiv subjects

Krešimir Josić

Publications and source records attributed to Krešimir Josić.

28 records · Page 2Linked to original sources

arXiv2016

Effects of cell cycle noise on excitable gene circuits

We assess the impact of cell cycle noise on gene circuit dynamics. For bistable genetic switches and excitable circuits, we find that transitions between metastable states most likely occur just after cell division and that this concentration effect intensifies in the presence of transcriptional delay. We explain this concentration effect with a 3-states stochastic model. For genetic oscillators, we quantify the temporal correlations between daughter cells induced by cell division. Temporal correlations must be captured properly in order to accurately quantify noise sources within gene networks.

q-bio.MN↗

arXiv2014

The validity of quasi steady-state approximations in discrete stochastic simulations

In biochemical networks, reactions often occur on disparate timescales and can be characterized as either "fast" or "slow." The quasi-steady state approximation (QSSA) utilizes timescale separation to project models of biochemical networks onto lower-dimensional slow manifolds. As a result, fast elementary reactions are not modeled explicitly, and their effect is captured by non-elementary reaction rate functions (e.g. Hill functions). The accuracy of the QSSA applied to deterministic systems depends on how well timescales are separated. Recently, it has been proposed to use the non-elementary rate functions obtained via the deterministic QSSA to define propensity functions in stochastic simulations of biochemical networks. In this approach, termed the stochastic QSSA, fast reactions that are part of non-elementary reactions are not simulated, greatly reducing computation time. However, it is unclear when the stochastic QSSA provides an accurate approximation of the original stochastic simulation. We show that, unlike the deterministic QSSA, the validity of the stochastic QSSA does not follow from timescale separation alone, but also depends on the sensitivity of the non-elementary reaction rate functions to changes in the slow species. The stochastic QSSA becomes more accurate when this sensitivity is small. Different types of QSSAs result in non-elementary functions with different sensitivities, and the total QSSA results in less sensitive functions than the standard or the pre-factor QSSA. We prove that, as a result, the stochastic QSSA becomes more accurate when non-elementary reaction functions are obtained using the total QSSA. Our work provides a novel condition for the validity of the QSSA in stochastic simulations of biochemical reaction networks with disparate timescales.

q-bio.MN↗

arXiv2014

Molecular mechanisms that regulate the coupled period of the mammalian circadian clock

In mammals, most cells in the brain and peripheral tissues generate circadian (~24hr) rhythms autonomously. These self-sustained rhythms are coordinated and entrained by a master circadian clock in the suprachiasmatic nucleus (SCN). Within the SCN, the individual rhythms of each neuron are synchronized through intercellular signaling. One important feature of SCN is that the synchronized period is close to the cell population mean of intrinsic periods. In this way, the synchronized period of the SCN stays close to the periods of cells in peripheral tissues. This is important for SCN to entrain cells throughout the body. However, the mechanism that drives the period of the coupled SCN cells to the population mean is not known. We use mathematical modeling and analysis to show that the mechanism of transcription repression plays a pivotal role in regulating the coupled period. Specifically, we use phase response curve analysis to show that the coupled period within the SCN stays near the population mean if transcriptional repression occurs via protein sequestration. In contrast, the coupled period is far from the mean if repression occurs through highly nonlinear Hill-type regulation (e.g. oligomer- or phosphorylation-based repression). Furthermore, we find that the timescale of intercellular coupling needs to be fast compared to that of intracellular feedback to maintain the mean period. These findings reveal the important relationship between the intracellular transcriptional feedback loop and intercellular coupling. This relationship explains why transcriptional repression appears to occur via protein sequestration in multicellular organisms, mammals and Drosophila, in contrast with the phosphorylation-based repression in unicellular organisms. That is, transition to protein sequestration is essential for synchronizing multiple cells with a period close to the population mean (~24hr).

q-bio.MN↗

arXiv2014

Modeling delay in genetic networks: From delay birth-death processes to delay stochastic differential equations

Delay is an important and ubiquitous aspect of many biochemical processes. For example, delay plays a central role in the dynamics of genetic regulatory networks as it stems from the sequential assembly of first mRNA and then protein. Genetic regulatory networks are therefore frequently modeled as stochastic birth-death processes with delay. Here we examine the relationship between delay birth-death processes and their appropriate approximating delay chemical Langevin equations. We prove that the distance between these two descriptions, as measured by expectations of functionals of the processes, converges to zero with increasing system size. Further, we prove that the delay birth-death process converges to the thermodynamic limit as system size tends to infinity. Our results hold for both fixed delay and distributed delay. Simulations demonstrate that the delay chemical Langevin approximation is accurate even at moderate system sizes. It captures dynamical features such as the spatial and temporal distributions of transition pathways in metastable systems, oscillatory behavior in negative feedback circuits, and cross-correlations between nodes in a network. Overall, these results provide a foundation for using delay stochastic differential equations to approximate the dynamics of birth-death processes with delay.

q-bio.MN↗

arXiv2013

Local paths to global coherence: cutting networks down to size

How does connectivity impact network dynamics? We address this question by linking network characteristics on two scales. On the global scale we consider the coherence of overall network dynamics. We show that such \emph{global coherence} in activity can often be predicted from the \emph{local structure} of the network. To characterize local network structure we use "motif cumulants," a measure of the deviation of pathway counts from those expected in a minimal probabilistic network model. We extend previous results in three ways. First, we give a new combinatorial formulation of motif cumulants that relates to the allied concept in probability theory. Second, we show that the link between global network dynamics and local network architecture is strongly affected by heterogeneity in network connectivity. However, we introduce a network-partitioning method that recovers a tight relationship between architecture and dynamics. Third, for a particular set of models we generalize the underlying theory to treat dynamical coherence at arbitrary orders (i.e. triplet correlations, and beyond). We show that at any order only a highly restricted set of motifs impact dynamical correlations.

q-bio.NC↗

arXiv2013

A generative spike train model with time-structured higher order correlations

Emerging technologies are revealing the spiking activity in ever larger neural ensembles. Frequently, this spiking is far from independent, with correlations in the spike times of different cells. Understanding how such correlations impact the dynamics and function of neural ensembles remains an important open problem. Here we describe a new, generative model for correlated spike trains that can exhibit many of the features observed in data. Extending prior work in mathematical finance, this generalized thinning and shift (GTaS) model creates marginally Poisson spike trains with diverse temporal correlation structures. We give several examples which highlight the model's flexibility and utility. For instance, we use it to examine how a neural network responds to highly structured patterns of inputs. We then show that the GTaS model is analytically tractable, and derive cumulant densities of all orders in terms of model parameters. The GTaS framework can therefore be an important tool in the experimental and theoretical exploration of neural dynamics.

q-bio.NC↗

arXiv2012

Piecewise linear models of chemical reaction networks

We show that certain non-linear dynamical systems with non-linearities in the form of Hill functions, can be approximated by piecewise linear dynamical systems. The resulting piecewise systems have closed form solutions that can be used to understand the behavior of the fully nonlinear system. We justify the reduction using geometric singular perturbation theory, and illustrate the results in networks modeling a genetic switch and a genetic oscillator.

q-bio.MN↗

arXiv2011

Impact of network structure and cellular response on spike time correlations

Novel experimental techniques reveal the simultaneous activity of larger and larger numbers of neurons. As a result there is increasing interest in the structure of cooperative -- or correlated -- activity in neural populations, and in the possible impact of such correlations on the neural code. A fundamental theoretical challenge is to understand how the architecture of network connectivity along with the dynamical properties of single cells shape the magnitude and timescale of correlations. We provide a general approach to this problem by extending prior techniques based on linear response theory. We consider networks of general integrate-and-fire cells with arbitrary architecture, and provide explicit expressions for the approximate cross-correlation between constituent cells. These correlations depend strongly on the operating point (input mean and variance) of the neurons, even when connectivity is fixed. Moreover, the approximations admit an expansion in powers of the matrices that describe the network architecture. This expansion can be readily interpreted in terms of paths between different cells. We apply our results to large excitatory-inhibitory networks, and demonstrate first how precise balance --- or lack thereof --- between the strengths and timescales of excitatory and inhibitory synapses is reflected in the overall correlation structure of the network. We then derive explicit expressions for the average correlation structure in randomly connected networks. These expressions help to identify the important factors that shape coordinated neural activity in such networks.

q-bio.NC↗

arXiv2011

Reduced models of networks of coupled enzymatic reactions

The Michaelis-Menten equation has played a central role in our understanding of biochemical processes. It has long been understood how this equation approximates the dynamics of irreversible enzymatic reactions. However, a similar approximation in the case of networks, where the product of one reaction can act as an enzyme in another, has not been fully developed. Here we rigorously derive such an approximation in a class of coupled enzymatic networks where the individual interactions are of Michaelis-Menten type. We show that the sufficient conditions for the validity of the total quasi steady state assumption (tQSSA), obtained in a single protein case by Borghans, de Boer and Segel can be extended to sufficient conditions for the validity of the tQSSA in a large class of enzymatic networks. Secondly, we derive reduced equations that approximate the network's dynamics and involve only protein concentrations. This significantly reduces the number of equations necessary to model such systems. We prove the validity of this approximation using geometric singular perturbation theory and results about matrix differentiation. The ideas used in deriving the approximating equations are quite general, and can be used to systematize other model reductions.

math-ph↗

arXiv2005

Synchronization in networks of general, weakly nonlinear oscillators

We present a general approach to the study of synchrony in networks of weakly nonlinear systems described by singularly perturbed equations of the type $x''+x+εf(x,x')=0$. By performing a perturbative calculation based on normal form theory we analytically obtain an $Ø(ε)$ approximation to the Floquet multipliers that determine the stability of the synchronous solution. The technique allows us to prove and generalize recent results obtained using heuristic approaches, as well as reveal the structure of the approximating equations. We illustrate the results in several examples, and discuss extensions to the analysis of stability of multisynchronous states in networks with complex architectures.

nlin.PS↗