A stochastic switch with different phases.

We describe an analog stochastic switch that exhibits three distinct phases as its parameters change. The phases are classified by the mean and variance of the switch's output. A phase change appears if the mean or the variance tends to a finite value or to infinity. The switch can be embedded in a large gene regulatory network for which the moment equations naturally close at the second order. This switch was used to model the response of a heat-shock system.

Hill functions for stochastic gene regulatory networks from master equations with split nodes and time-scale separation.

The deterministic Hill function depends only on the average values of molecule numbers. To account for the fluctuations in the molecule numbers, the argument of the Hill function needs to contain the means, the standard deviations, and the correlations. Here we present a method that allows for stochastic Hill functions to be constructed from the dynamical evolution of stochastic biocircuits with specific topologies. These stochastic Hill functions are presented in a closed analytical form so that they can be easily incorporated in models for large genetic regulatory networks. Using a repressive biocircuit as an example, we show by Monte Carlo simulations that the traditional deterministic Hill function inaccurately predicts time of repression by an order of two magnitudes. However, the stochastic Hill function was able to capture the fluctuations and thus accurately predicted the time of repression.

Splitting nodes and linking channels: A method for assembling biocircuits from stochastic elementary units.

Akin to electric circuits, we construct biocircuits that are manipulated by cutting and assembling channels through which stochastic information flows. This diagrammatic manipulation allows us to create a method which constructs networks by joining building blocks selected so that (a) they cover only basic processes; (b) it is scalable to large networks; (c) the mean and variance-covariance from the Pauli master equation form a closed system; and (d) given the initial probability distribution, no special boundary conditions are necessary to solve the master equation. The method aims to help with both designing new synthetic signaling pathways and quantifying naturally existing regulatory networks.

A Kramers-Moyal approach to the analysis of third-order noise with applications in option valuation.

We propose the use of the Kramers-Moyal expansion in the analysis of third-order noise. In particular, we show how the approach can be applied in the theoretical study of option valuation. Despite Pawula's theorem, which states that a truncated model may exhibit poor statistical properties, we show that for a third-order Kramers-Moyal truncation model of an option's and its underlier's price, important properties emerge: (i) the option price can be written in a closed analytical form that involves the Airy function, (ii) the price is a positive function for positive skewness in the distribution, (iii) for negative skewness, the price becomes negative only for price values that are close to zero. Moreover, using third-order noise in option valuation reveals additional properties: (iv) the inconsistencies between two popular option pricing approaches (using a "delta-hedged" portfolio and using an option replicating portfolio) that are otherwise equivalent up to the second moment, (v) the ability to develop a measure R of how accurately an option can be replicated by a mixture of the underlying stocks and cash, (vi) further limitations of second-order models revealed by introducing third-order noise.

Sensing the heat stress by Mammalian cells.

BACKGROUND: The heat-shock response network controls the adaptation and survival of the cell against environmental stress. This network is highly conserved and is connected with many other signaling pathways. A key element of the heat-shock network is the heat-shock transcription factor-1 (HSF), which is transiently activated by elevated temperatures. HSF translocates to the nucleus upon elevated temperatures, forming homotrimeric complexes. The HSF homotrimers bind to the heat shock element on the DNA and control the expression of the hsp70 gene. The Hsp70 proteins protect cells from thermal stress. Thermal stress causes the unfolding of proteins, perturbing thus the pathways under their control. By binding to these proteins, Hsp70 allows them to refold and prevents their aggregation. The modulation of the activity of the hsp70-promoter by the intensity of the input stress is thus critical for cell's survival. The promoter activity starts from a basal level and rapidly increases once the stress is applied, reaches a maximum level and attenuates slowely back to the basal level. This phenomenon is the hallmark of many experimental studies and of all computational network analysis. RESULTS: The molecular construct used as a measure of the response to thermal stress is a Hsp70-GFP fusion gene transfected in Chinese hamster ovary (CHO) cells. The time profile of the GFP protein depends on the transient activity, Transient(t), of the heat shock system. The function Transient(t) depends on hsp70 promoter activity, transcriptional regulation and the translation initiation effects elicited by the heat stress. The GFP time profile is recorded using flow cytometry measurements, a technique that allows a quantitative measurement of the fluorescence of a large number of cells (104). The GFP responses to one and two heat shocks were measured for 261 conditions of different temperatures and durations. We found that: (i) the response of the cell to two consecutive shocks (i.e., no recovery time in between shocks) depends on the order of the input shocks, that is the shocks do not commute; (ii) the responses may be classified as mild or severe, depending on the temperature level and the duration of the heat shock and (iii) the response is highly sensitive to small variations in temperature. CONCLUSIONS: We propose a mathematical model that maps temperature into the transient activity using experimental data that describes the time course of the response to input thermal stress. The model is built on thermotolerance without recovery time, sharp sensitivity to small variations in temperature and the existence of mild and severe classes of stress responses. The theoretical predictions are tested against experimental data using a series of double-shock inputs. The theoretical structure is represented by a sequence of three cascade processes that transform the input stress into the transient activity. The structure of the cascade is nonlinear-linear-nonlinear (NLN). The first nonlinear system (N) from the NLN structure represents the amplification of small changes in the environmental temperature; the linear system (L) represents the thermotolerance without recovery time, whereas the last system (N) represents the transition of the cell's response from a mild to a severe shock.

The effect of coupled stochastic processes in a two-state biochemical switch.

Cell signaling pathways consist of multiple connections of different types of gene, mRNA and protein networks. It is not a trivial task to follow the signals flowing through these networks. The difficulty comes from considering the entire biological structure as a single network without breaking it into connected modules. The study of these networks simplifies if the complex system is reduced to a hierarchy of interconnected modules. Out of many potential modules, a specific one, namely the Goldbeter-Koshland switch, was encountered by the authors during their study of the Mammalian Heat Shock Response Network (MHSRN) where the switch acts as a stress sensor. Usually, only the steady state behavior of the switch is studied, in which the phosphorylated protein is given as a function of the enzyme concentration. Experimental results show that the heat shock response is still present 20 h after the temperature stress had ended. Thus, it is useful to analyze the transient behavior of the switch that couples the environment to the MHSRN. A stochastic model for the switch is proposed using the Master Equation which is subsequently transformed into an equation for the factorial cumulant generating function. This generating function can be easily read from a graphical representation of the stochastic switch. The second order approximation of the equation for the factorial cumulant generating function is solved and the time dependence of the transient regime of the mean and standard deviation is readily obtained. Using the mean and standard deviation of the switch's output as a function of the stochastic input signals that represent the environment, we classify the switches according to different criteria. The switches differ by the numerical values of the parameters that characterize the switch's chemical reactions. The classifying criteria will distinguish the switches by the levels of the response for a given transition time and by the sensitivity of the response to the enzyme levels. It is also found that the environment can drastically change the response of the switch, which has important biological consequences.

Impact of Interdisciplinary Undergraduate Research in mathematics and biology on the development of a new course integrating five STEM disciplines.

Funded by innovative programs at the National Science Foundation and the Howard Hughes Medical Institute, University of Richmond faculty in biology, chemistry, mathematics, physics, and computer science teamed up to offer first- and second-year students the opportunity to contribute to vibrant, interdisciplinary research projects. The result was not only good science but also good science that motivated and informed course development. Here, we describe four recent undergraduate research projects involving students and faculty in biology, physics, mathematics, and computer science and how each contributed in significant ways to the conception and implementation of our new Integrated Quantitative Science course, a course for first-year students that integrates the material in the first course of the major in each of biology, chemistry, mathematics, computer science, and physics.

Heat shock response in CHO mammalian cells is controlled by a nonlinear stochastic process.

In many biological systems, the interactions that describe the coupling between different units in a genetic network are nonlinear and stochastic. We study the interplay between stochasticity and nonlinearity using the responses of Chinese hamster ovary (CHO) mammalian cells to different temperature shocks. The experimental data show that the mean value response of a cell population can be described by a mathematical expression (empirical law) which is valid for a large range of heat shock conditions. A nonlinear stochastic theoretical model was developed that explains the empirical law for the mean response. Moreover, the theoretical model predicts a specific biological probability distribution of responses for a cell population. The prediction was experimentally confirmed by measurements at the single-cell level. The computational approach can be used to study other nonlinear stochastic biological phenomena.

Is the future biology Shakespearean or Newtonian?

"Cells do not care about mathematics" thus concluded a biologist friend after a discussion on the future of biology. And indeed, why should they care? But if we exchange the word "cell" with "rock", "Moon" or "electrons", do we have to change the sentence also? Starting from this line of thought, we review some recent developments in understanding the stochastic behavior of biological systems. We emphasize the importance of a molecular Signal Generator in the study of genetic networks.

The use of oscillatory signals in the study of genetic networks.

The structure of a genetic network is uncovered by studying its response to external stimuli (input signals). We present a theory of propagation of an input signal through a linear stochastic genetic network. We found that there are important advantages in using oscillatory signals over step or impulse signals and that the system may enter into a pure fluctuation resonance for a specific input frequency.

Distinct temporal genetic signatures of neurogenic and gliogenic cues in cortical stem cell cultures.

Cortical progenitor cells from rat embryos give rise to neurons or glia following exposure to platelet derived growth factor (PDGF) or ciliary neurotrophic factor (CNTF), respectively. Both growth factors impart their developmental cues quickly through a transcription-dependent mechanism. Do the alternate developmental responses to PDGF and CNTF reflect induction of qualitatively distinct genes? Alternatively, do the same genes respond to each growth factor, but with quantitatively distinct kinetics? Using differential library screening and custom cDNA microarrays we show that a common set of genes responds to either growth factor. However, quantitative differences in the onset and duration of gene induction equate to the expression of factor-specific gene signatures. Multitissue cluster analysis also reveals tissue-specific gene signatures that may play important roles in the developing brain.

GoSurfer: a graphical interactive tool for comparative analysis of large gene sets in Gene Ontology space.

UNLABELLED: The analysis of complex patterns of gene regulation is central to understanding the biology of cells, tissues and organisms. Patterns of gene regulation pertaining to specific biological processes can be revealed by a variety of experimental strategies, particularly microarrays and other highly parallel methods, which generate large datasets linking many genes. Although methods for detecting gene expression have improved substantially in recent years, understanding the physiological implications of complex patterns in gene expression data is a major challenge. This article presents GoSurfer, an easy-to-use graphical exploration tool with built-in statistical features that allow a rapid assessment of the biological functions represented in large gene sets. GoSurfer takes one or two list(s) of gene identifiers (Affymetrix probe set ID) as input and retrieves all the Gene Ontology (GO) terms associated with the input genes. GoSurfer visualises these GO terms in a hierarchical tree format. With GoSurfer, users can perform statistical tests to search for the GO terms that are enriched in the annotations of the input genes. These GO terms can be highlighted on the GO tree. Users can manipulate the GO tree in various ways and interactively query the genes associated with any GO term. The user-generated graphics can be saved as graphics files, and all the GO information related to the input genes can be exported as text files. AVAILABILITY: GoSurfer is a Windows-based program freely available for noncommercial use and can be downloaded at http://www.gosurfer.org. Datasets used to construct the trees shown in the figures in this article are available at http://www.gosurfer.org/download/GoSurfer.zip.

Extensive and divergent circadian gene expression in liver and heart.

Many mammalian peripheral tissues have circadian clocks; endogenous oscillators that generate transcriptional rhythms thought to be important for the daily timing of physiological processes. The extent of circadian gene regulation in peripheral tissues is unclear, and to what degree circadian regulation in different tissues involves common or specialized pathways is unknown. Here we report a comparative analysis of circadian gene expression in vivo in mouse liver and heart using oligonucleotide arrays representing 12,488 genes. We find that peripheral circadian gene regulation is extensive (> or = 8-10% of the genes expressed in each tissue), that the distributions of circadian phases in the two tissues are markedly different, and that very few genes show circadian regulation in both tissues. This specificity of circadian regulation cannot be accounted for by tissue-specific gene expression. Despite this divergence, the clock-regulated genes in liver and heart participate in overlapping, extremely diverse processes. A core set of 37 genes with similar circadian regulation in both tissues includes candidates for new clock genes and output genes, and it contains genes responsive to circulating factors with circadian or diurnal rhythms.