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1.  A Kramers-Moyal Approach to the Analysis of Third-Order Noise with Applications in Option Valuation 
PLoS ONE  2015;10(1):e0116752.
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.
PMCID: PMC4308111  PMID: 25625856
2.  The effect of coupled stochastic processes in a two-state biochemical switch 
Journal of Biological Physics  2011;37(4):441-462.
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.
PMCID: PMC3169695  PMID: 22942487
Goldbeter–Koshland switch; Stochastic; Heat shock
3.  Sensing the Heat Stress by Mammalian Cells 
BMC Biophysics  2011;4:16.
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.
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.
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.
PMCID: PMC3180696  PMID: 21834999
4.  Impact of Interdisciplinary Undergraduate Research in Mathematics and Biology on the Development of a New Course Integrating Five STEM Disciplines 
CBE Life Sciences Education  2010;9(3):212-216.
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.
PMCID: PMC2931668  PMID: 20810953
5.  Heat Shock Response in CHO Mammalian Cells Is Controlled by a Nonlinear Stochastic Process 
PLoS Computational Biology  2007;3(10):e187.
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.
Author Summary
The structure of an unknown biological system is uncovered by experimentally perturbing the system with a series of input signals. The response to these perturbations is measured as output signals. Then, the mathematical relation between the input and the output signals constitutes a model for the system. As a result, a classification of biological molecular networks can be devised using their input–output functional relation. This article studies the input–output functional form for the response to heat shocks in mammalian cells. The Chinese hamster ovary (CHO) mammalian cells were perturbed with a series of heat pulses of precise duration and temperature. The experimental data, taken at the single-cell level, revealed a simple and precise mathematical law for the time evolution of the heat shock response. Parameters of the mathematical law can be experimentally measured and can be used by heat shock biologists to classify the heat shock response in different experimental conditions. Since the response to heat shock is the outcome of a transcriptional factor control, it is highly probable that the empirical law is valid for other biological systems. The mathematical model explains not only the mean value of the response but also the time evolution of its probability distribution in a cell population.
PMCID: PMC2000973  PMID: 17922567

Results 1-5 (5)