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**1 - 9**of**9**### Entitled: Consistency and Sensitivity Analysis of Multi-Level Petri Net Models of Biological Systems

, 2013

"... complies with the regulations of the University and meets the accepted standards with respect to originality and quality. Signed by the final examining committee: Chair ..."

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complies with the regulations of the University and meets the accepted standards with respect to originality and quality. Signed by the final examining committee: Chair

### Characterization of Reachable Attractors Using Petri Net Unfoldings

"... Abstract. Attractors of network dynamics represent the long-term be-haviours of the modelled system. Their characterization is therefore cru-cial for understanding the response and differentiation capabilities of a dynamical system. In the scope of qualitative models of interaction net-works, the co ..."

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Abstract. Attractors of network dynamics represent the long-term be-haviours of the modelled system. Their characterization is therefore cru-cial for understanding the response and differentiation capabilities of a dynamical system. In the scope of qualitative models of interaction net-works, the computation of attractors reachable from a given state of the network faces combinatorial issues due to the state space explosion. In this paper, we present a new algorithm that exploits the concurrency between transitions of parallel acting components in order to reduce the search space. The algorithm relies on Petri net unfoldings that can be used to compute a compact representation of the dynamics. We illustrate the applicability of the algorithm with Petri net models of cell signalling and regulation networks, Boolean and multi-valued. The proposed ap-proach aims at being complementary to existing methods for deriving the attractors of Boolean models, while being generic since it applies to any safe Petri net.

### and Manufacturing Innovation STRUCTURE AND DYNAMICS OF POLYNOMIAL DYNAMICAL SYSTEMS ∗

"... Discrete models have a long tradition in engi-neering, including finite state machines, Boolean networks, Petri nets, and agent-based models. Of particular importance is the question of how the model structure constrains its dynamics. This paper discusses an algebraic framework to study such questio ..."

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Discrete models have a long tradition in engi-neering, including finite state machines, Boolean networks, Petri nets, and agent-based models. Of particular importance is the question of how the model structure constrains its dynamics. This paper discusses an algebraic framework to study such questions. The systems discussed here are given by mappings on an affine space over a finite field, whose coordinate functions are polynomials. They form a general class of models which can represent many discrete model types. Assigning to such a system its depen-dency graph, that is, the directed graph that indicates the variable dependencies, provides a mapping from systems to graphs. A basic prop-erty of this mapping is derived and used to prove that dynamical systems with an acyclic depen-dency graph can only have a unique fixed point in their phase space and no periodic orbits. This result is then applied to a published model of in vitro virus competition. ∗This work was supported by the National Science Foundation under Grant Nr. CMMI-0908201.

### BOOLEAN NETWORKS WITH MULTI-EXPRESSIONS AND PARAMETERS

"... Abstract. To model biological systems using networks, it is de-sirable to allow more than two levels of expression for the nodes and to allow the introduction of parameters. Various modeling and simulation methods addressing these needs using Boolean models, both synchronous and asynchronous, have b ..."

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Abstract. To model biological systems using networks, it is de-sirable to allow more than two levels of expression for the nodes and to allow the introduction of parameters. Various modeling and simulation methods addressing these needs using Boolean models, both synchronous and asynchronous, have been proposed in the lit-erature. However, analytical study of these more general Boolean networks models is lagging. This paper aims to develop a concise theory for these different Boolean logic based modeling methods. Boolean models for networks where each node can have more than two levels of expression and Boolean models with parameters are defined algebraically with examples provided. Certain classes of random asynchronous Boolean networks and deterministic mod-uli asynchronous Boolean networks are investigated in detail using the setting introduced in this paper. The derived theorems provide a clear picture for the attractor structures of these asynchronous Boolean networks. 1.

### Multi-level Modeling of Light-Induced Stomatal Opening Offers New Insights into Its Regulation by Drought

"... Plant guard cells gate CO2 uptake and transpirational water loss through stomatal pores. As a result of decades of experimental investigation, there is an abundance of information on the involvement of specific proteins and secondary messengers in the regulation of stomatal movements and on the pair ..."

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Plant guard cells gate CO2 uptake and transpirational water loss through stomatal pores. As a result of decades of experimental investigation, there is an abundance of information on the involvement of specific proteins and secondary messengers in the regulation of stomatal movements and on the pairwise relationships between guard cell components. We constructed a multi-level dynamic model of guard cell signal transduction during light-induced stomatal opening and of the effect of the plant hormone abscisic acid (ABA) on this process. The model integrates into a coherent network the direct and indirect biological evidence regarding the regulation of seventy components implicated in stomatal opening. Analysis of this signal transduction network identified robust cross-talk between blue light and ABA, in which [Ca2+]c plays a key role, and indicated an absence of cross-talk between red light and ABA. The dynamic model captured more than 1031 distinct states for the system and yielded outcomes that were in qualitative agreement with a wide variety of previous experimental results. We obtained novel model predictions by simulating single component knockout phenotypes. We found that under white light or blue light, over 60%, and under red light, over 90 % of all simulated knockouts had similar opening responses as wild type, showing that the system is robust against single node loss. The model revealed an open question concerning the effect of ABA on red light-induced stomatal opening. We experimentally showed that ABA is able to inhibit red light-induced stomatal opening, and our model offers possible hypotheses for the underlying mechanism, which point to

### 1A Network Biology Approach to Denitrification in Pseudomonas aeruginosa

"... Pseudomonas aeruginosa is a metabolically flexible member of the Gammaproteobacteria. Under anaer-obic conditions and the presence of nitrate, P. aeruginosa can perform (complete) denitrification, a respiratory process of dissimilatory nitrate reduction to nitrogen gas via nitrite (NO2), nitric oxid ..."

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Pseudomonas aeruginosa is a metabolically flexible member of the Gammaproteobacteria. Under anaer-obic conditions and the presence of nitrate, P. aeruginosa can perform (complete) denitrification, a respiratory process of dissimilatory nitrate reduction to nitrogen gas via nitrite (NO2), nitric oxide (NO) and nitrous oxide (N2O). This study focuses on understanding the influence of environmental condi-tions on bacterial denitrification performance, using a mathematical model of a metabolic network in P. aeruginosa. To our knowledge, this is the first mathematical model of denitrification for this bacterium. Analysis of the long-term behavior of the network under changing concentration levels of oxygen (O2), nitrate (NO3), and phosphate (PO4) suggests that PO4 concentration strongly affects denitrification performance. The model provides three predictions on denitrification activity of P. aeruginosa under various environmental conditions, and these predictions are either experimentally validated or supported by pertinent biological literature. One motivation for this study is to capture the effect of PO4 on a denitrification metabolic network of P. aeruginosa in order to shed light on mechanisms for greenhouse gas N2O accumulation during seasonal oxygen depletion in aquatic environments such as Lake Erie (Laurentian Great Lakes, USA).

### On the Feasibility of MapReduce to Compute Phase Space Properties of Graphical Dynamical Systems: An Empirical Study

, 2015

"... A graph dynamical system (GDS) is a theoretical construct that can be used to simulate and analyze the dynamics of contagion processes on network representations of systems. One of our goals is to compute the phase space of a system (i.e., the set of all transitions for each state of the system), an ..."

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A graph dynamical system (GDS) is a theoretical construct that can be used to simulate and analyze the dynamics of contagion processes on network representations of systems. One of our goals is to compute the phase space of a system (i.e., the set of all transitions for each state of the system), and for this, even 30-vertex graphs present a computational challenge. This is because the number of states is exponential in the number of graph vertices. We implement several algorithms for phase space computations based on naive approaches, and based on more recent graph dynamical systems theory. We find that MapReduce is not well suited for these types of computations. While we observe improvements in execution times with algorithmic improvements, these improvements are not sufficient to overcome inefficiencies of MapReduce for these types of computations. We present a detailed set of observations explaining system performance. Future work entails executing our algorithms on Apache Spark. Acknowledgments First and foremost, I thank my advisor, Dr. Madhav V. Marathe, for giving me the wonderful

### RESEARCH ARTICLE GDSCalc: A Web-Based Application for Evaluating Discrete Graph Dynamical Systems

"... Discrete dynamical systems are used to model various realistic systems in network science, from social unrest in human populations to regulation in biological networks. A common approach is to model the agents of a system as vertices of a graph, and the pairwise interac-tions between agents as edges ..."

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Discrete dynamical systems are used to model various realistic systems in network science, from social unrest in human populations to regulation in biological networks. A common approach is to model the agents of a system as vertices of a graph, and the pairwise interac-tions between agents as edges. Agents are in one of a finite set of states at each discrete time step and are assigned functions that describe how their states change based on neigh-borhood relations. Full characterization of state transitions of one system can give insights into fundamental behaviors of other dynamical systems. In this paper, we describe a dis-crete graph dynamical systems (GDSs) application called GDSCalc for computing and characterizing system dynamics. It is an open access system that is used through a web interface. We provide an overview of GDS theory. This theory is the basis of the web appli-cation; i.e., an understanding of GDS provides an understanding of the software features, while abstracting away implementation details. We present a set of illustrative examples to demonstrate its use in education and research. Finally, we compare GDSCalc with other discrete dynamical system software tools. Our perspective is that no single software tool will perform all computations that may be required by all users; tools typically have particular features that are more suitable for some tasks. We situate GDSCalc within this space of software tools.

### RESEARCH ARTICLE A Network Biology Approach to Denitrification in Pseudomonas aeruginosa

"... Pseudomonas aeruginosa is a metabolically flexible member of the Gammaproteobacteria. Under anaerobic conditions and the presence of nitrate, P. aeruginosa can perform (com-plete) denitrification, a respiratory process of dissimilatory nitrate reduction to nitrogen gas via nitrite (NO2), nitric oxid ..."

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Pseudomonas aeruginosa is a metabolically flexible member of the Gammaproteobacteria. Under anaerobic conditions and the presence of nitrate, P. aeruginosa can perform (com-plete) denitrification, a respiratory process of dissimilatory nitrate reduction to nitrogen gas via nitrite (NO2), nitric oxide (NO) and nitrous oxide (N2O). This study focuses on under-standing the influence of environmental conditions on bacterial denitrification performance, using a mathematical model of a metabolic network in P. aeruginosa. To our knowledge, this is the firstmathematical model of denitrification for this bacterium. Analysis of the long-term behavior of the network under changing concentration levels of oxygen (O2), nitrate (NO3), and phosphate (PO4) suggests that PO4 concentration strongly affects denitrification performance. The model provides three predictions on denitrification activity of P. aerugi-nosa under various environmental conditions, and these predictions are either experimen-tally validated or supported by pertinent biological literature. One motivation for this study is to capture the effect of PO4 on a denitrification metabolic network of P. aeruginosa in order to shed light on mechanisms for greenhouse gas N2O accumulation during seasonal oxy-gen depletion in aquatic environments such as Lake Erie (Laurentian Great Lakes, USA). Simulating the microbial production of greenhouse gases in anaerobic aquatic systems such as Lake Erie allows a deeper understanding of the contributing environmental effects that will inform studies on, and remediation strategies for, other hypoxic sites worldwide.