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.Comment: 21 pages, 4 figure

Resource Allocation Techniques for Wireless Powered Communication Networks with Energy Storage Constraint

This paper studies multi-user wireless powered communication networks, where energy constrained users charge their energy storages by scavenging energy of the radio frequency signals radiated from a hybrid access point (H-AP). The energy is then utilized for the users' uplink information transmission to the H-AP in time division multiple access mode. In this system, we aim to maximize the uplink sum rate performance by jointly optimizing energy and time resource allocation for multiple users in both infinite capacity and finite capacity energy storage cases. First, when the users are equipped with the infinite capacity energy storages, we derive the optimal downlink energy transmission policy at the H-AP. Based on this result, analytical resource allocation solutions are obtained. Next, we propose the optimal energy and time allocation algorithm for the case where each user has finite capacity energy storage. Simulation results confirm that the proposed algorithms offer 30% average sum rate performance gain over conventional schemes

Mathematical Modeling and Analysis of Cellular Clocks.

Cells generate various biological rhythms that control important aspects of cell physiology including circadian (daily) events, cell division, embryogenesis, DNA damage repair and metabolism. Since these cellular rhythms can determine the fitness or fate of organisms, how cells generate and control rhythms has become a central problem in biology. In this dissertation, we have developed theorems and mathematical models to understand how complex biochemical interactions of many genes and proteins generate and control biological rhythms over a wide range of conditions. In chapter 2, we have developed a mathematical theory that can infer biochemical interaction network of cellular clocks from timecourse data of gene and protein expression, which are relatively easy to be measured with the recent advances in experimental technology. We formulated this question as an existence and uniqueness problem and proved that the biochemical interaction network, and even biochemical rates, can sometimes uniquely be determined from only gene and protein timecourses. This theory provides a simple algorithm to determine whether two given species have a biochemical interaction. In chapter 3, we have found how cells generate rhythms with a constant period over a wide range of environmental conditions by studying circadian rhythms whose 24hr period is tightly regulated. By developing the most detailed and accurate mathematical model of circadian clock to date, we found that balancing a 1-1 stoichiometry between activators and repressors via double negative feedback loops is a key mechanism that tightly regulates the period of circadian rhythms. This mechanism provides an explanation for why various types of circadian disorders fail to regulate rhythms. In chapter 4, we considered rhythms of p53, one of the most important tumor suppressors. Unlike self-sustained circadian rhythms, p53 rhythms only occur in response to external stimuli such as DNA damage. Sustaining p53 rhythms is essential for p53 to repair DNA damage. By developing a mathematical model of p53 rhythms, we found that additional positive feedback loops via Rora and Cyt-c can significantly improve the sustainability of p53 rhythms in the presence of genetic heterogeneity and stochasticity.PhDApplied and Interdisciplinary MathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/99957/1/jaekkim_1.pd

A mechanism for robust circadian timekeeping via stoichiometric balance

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/102189/1/msb201262.reviewer_comments.pdfhttp://deepblue.lib.umich.edu/bitstream/2027.42/102189/2/msb201262.pdfhttp://deepblue.lib.umich.edu/bitstream/2027.42/102189/3/msb201262-sup-0001.pd