2012 Workshop for Young Researchers in Mathematical Biology (WYRMB): Abstracts and Lecture Materials
Plenary Speakers
A variety of new methods in ecological statistics aim to estimate population densities, and other characteristics, when organisms sometimes go undetected. I will present three applications that address the same general question but come from an unusual perspective, emphasizing the fun and utility of applying basic probability theory to sampling in ecological systems.
The main mechanism of tumour growth is the surface diffusion of cells at tumour border. Only cells that can divide are those at the border, where there is still space available; indeed, space is being made constantly available by the lytic processes unleashed against the host tissue. Thus, while the cells at the border continue to grow, those within the tumour mass become quiescent and eventually necrotic. The importance of new cell movements lies in the fact that tumour growth must be conceived as a competition for space between the tumour and the host, and not for nutrients or other factors. An unexpected emergent behaviour of neutrophils arising from tumour growth dynamics is its capability to compete for space with tumour cells. Then, the immune innate response of the organism plays the key role in the fight of tumours. If the organism is able to send a number enough of neutrophils around tumour, the latter will regress and necrose. A powerful anti-tumoural barrier of neutrophils would block the potential space into which a tumour can grow, i.e., the cavities on the tumour border in which new, diffusing tumour cells settle. But, if the number of neutrophils is low, their presence may even help tumour growth. It should be remembered that neutrophils also have a degrading effect on the organ in which a tumour lies, so enough have to arrive for the tumour-stopping effect to outweigh this negative effect. Then a threshold number of neutrophils must exist if a tumour is to be beaten. This explains why immunosuppressed patients often develop tumours - they cannot mount sufficiently large neutrophil attacks against them when they appear. The hypothesis proposed here is therefore simple: ensuring the massive recruitment of neutrophils to the tumour border should successfully prevent tumour growth and lead to tumour involution. In this talk, a series of theoretical, experimental and clinical works are explained to fully understand and to support this hypothesis.
I will present a number of problems in macromolecular cellular, and tissue level biology that can be modeled using approaches from statistical physics, stochastic processes, and membrane mechanics. First, I will consider a simple model of nucleosome positioning that predicts the coverage of DNA by histones. At the cellular level, the evolution of cell populations can be described by nonequilibrium statistical mechanical models such as the zero-range process, with birth and death. We have applied this type of model to describe cancer progression. Finally, at the tissue level, basic membrane mechanics will be used to a mathematical framework for retinal detachments. These examples are meant to highlight the versatility of using basic paradigms from condensed matter and statistical physics to distill complex problems in cell biology and physiology.
Immunotherapy, a treatment approach that enhances the body's natural ability to fight cancers, is becoming increasingly prevalent in many multi-stage treatment programs that also include chemotherapy, radiation and surgery. The critical importance of the immune system in combating cancer has been verified clinically, as well as through mathematical models. However, many open questions remain regarding non-uniform patient responses to treatments, and how to optimize and personalize therapy protocols. Mathematical models can help to provide some insight into the mechanisms that may be influencing patient outcomes. A key to making progress in developing useful mathematical models of cancer-immunology dynamics is to work collaboratively across disciplinary boundaries. In this talk, we will present a variety of models and outcomes that have resulted from such interdisciplinary collaborations. We will discuss approaches to modeling cancer growth and immune system interactions, and treatment approaches that harness the power of the immune system to slow or even stop cancer progression.
I will discuss my approach to doing mathematical biology, which is by no means the best and hopefully not the worst, based on a simple rule: we have made a contribution when our collaborators say we have. * Thus far, I have developed four inspirational (for me) collaborations in math biology: a huge effort called the Virtual Lung Project; a study of single cell mechanochemical oscillations; a study of the yeast mitotic spindle in metaphase; and a study of viral-antibody interactions. I will discuss what I find cool about each of these projects, biologically and mathematically, and in particular why they are attractive for young mathematicians. For young researchers, it is important to know how to start, even more so how to sustain, a meaningful relationship and collaboration in math biology.
* A theme I borrowed from Fred Brooks, who started the Computer Science Department at UNC.
Computation is playing an ever increasing and vital role in the biological and healthcare sciences. In many instances, scientists are developing mathematical models and using high performance computing to carry out analysis and simulations that provide insight into biological systems. The complexity of these models often demands increasing compute power and sophisticated mathematics for the solution. This is coupled with increasing amounts of data that must be analyzed. High performance computing (HPC) is a tool frequently used to understand these complex problems involving large amounts of data in the life sciences. In this talk, I will give an overview of HPC directions and comment on the ever increasing amounts of data driving some of the computing directions. In conclusion through some illustrative examples, I will point out some of the computational trends that I believe hold opportunity for coupling high performance computing and mathematics to tackle life science problems.
The overarching goal of our research is to create quantitative tools that support tissue engineering efforts to predictably direct the differentiation, integration, and organization of living cells. To support these efforts, our research addresses model-based optimal experiment design and model-based design of control strategies. Experiments to help understand, resolve, and direct cellular processes are expensive. It is therefore vital to design experiments that will be nearly optimal among available experiments in terms of the information they reveal and their likelihood of success. Our approaches employ sparse grid methods to enable systematic and computationally efficient exploration over uncertain model parameter spaces with multiple potential model structures. We work in concert with collaborators to evaluate, refine, and extend our model-based experiment design and model-based control theory based approaches. The talk will reflect upon some of the challenges (and humorous incidents) in establishing a productive and rewarding collaboration between mathematicians, engineers, and life scientists.
Predicting changes in community composition and ecosystem function in a rapidly changing world is a major research challenge in ecology and evolution. I will discuss a proposed theoretical framework for addressing this challenge comprised of three elements: an underlying trait distribution (e.g., frequency distribution of photosynthetic rate across individuals and species in a community), a performance filter defining the fitness of traits in different environments, and a dynamic projection of the performance filter along some environmental gradient. This framework allows changes in the trait distribution and associated modifications to community composition or ecosystem function to be predicted across time or space. I will discuss analytical results using dynamical systems models within this framework that incorporate 1) migration from a global pool 2) an island model of migration and 3) correlations among traits and environmental drivers. These results help illustrate the underlying assumptions of traits-based models in the ecological literature and describe some biologically counter-intuitive results where lack of optimization (due to correlation) results in a faster evolutionary/ecological response in the trait distribution to environmental changes. Along with this analytical approach, I will also present an application of this framework to predicting species composition changes at Konza prairie using Bayesian hierarchical modeling, which helps to illustrate the difficulties in applying traits-based approaches to empirical data.
Short Talks
Consider the tasks of locating a friend in a crowd or finding a particular piece of paper in a pile. In these, and many other situations we need to find a target among distractors. The difficulty of such tasks depends on a number of factors. For instance, the task generally becomes more difficult as the number of distractors increases - typically it is harder to find a friend in large crowd. However, a friend wearing a red shirt will pop out in a crowd of people wearing blue. Hence the homogeneity among distractors is also important. Mazyar et al.(Nature Neuroscience, Vol.14, No.6, 2011) studied the impact of varying reliability of sensory information in the two extreme cases of completely heterogeneous and homogeneous distractors. They showed that humans adopt a near-optimal strategy in finding the target in both cases. They also found that when the distractors are homogeneous, the ability to find a target does not depend strongly on set size (Journal of Vision, 12(6):10, 1-16, 2012).
We study to what extent human observers take into account the homogeneity among distractors to detect a target. The subjects report whether a target stimulus (a vertically oriented ellipsoid) is present in a field of distractors (ellipsoids with non-vertical orientation). The variability in the orientation of distractors is correlated. In the extreme cases, the orientations of all distractors are equal (perfectly correlated), or chosen independently. However, we also consider examples where orientations are not identical, but dependent. Based on this data we ask whether humans are capable of learning the underlying correlation structure in the given visual scenes.
Many infectious diseases are caused by parasites and pathogens that are vectored by insects. The evolution of insecttransmitted parasites is shaped by interactions with both vertebrate and insect hosts. Pigeons have many parasites in the wild; however, our study focuses on two of these parasites: hippoboscid fly - the macroparasite and a malaria parasite: Haemoproteus columbae - the microparasite and their interactions with the pigeon and the interaction between them. Malaria in birds can be a serious parasitic disease, as it often is in humans. Some birds die from the infection while others spread it. Hippoboscid flies take their blood meals from pigeons, which are often infected with malaria. The fly then acts as a vector, transferring malaria between bird hosts. The malaria parasite must undergo a sexual reproductive stage in the fly and an asexual reproductive stage in the bird to complete its life cycle, thus potentially impacting the fitness of both the bird and the fly. Pigeons make antibodies to flies when exposed to biting supported by the experimental data which shows the change in antibody level, measured as "optical density". The birds with flies in their backpack have significantly greater changes in their fly-specific antibody levels when exposed to flies. As pigeons develop fly antibodies, this has an impact on the transmission of flies and consequently on the disease prevalence. Also the disease prevalence depending on the fly transmission has a feedback on the persistence of fly population.
We are investigating the system from two perspectives through mathematical modelling. From the parasitic fly's point of view we are interested in the effects of malaria on fly fitness. Understanding whether malaria impacts the fitness of its vector, has implications for the transmission dynamics of malaria and possibly other vectored pathogens. From the host's point of view we are interested in how hosts combat parasites immunologically. In this project we have seen host immunological defenses against vector affect vector transmission as well as its colonization with the host which in turn affects the disease prevalence and fly population size. This study has a resemblance with the vector borne diseases of human malaria. This is also relevant to understand the vector dynamics in disease transmission and implementing control strategies through anti-vector vaccines designed to target the vectors in such a way that they protect against vector feeding and so pathogens transmitted by the vector - which is a new approach.
Calcium plays a crucial role in a huge range of cellular processes including muscle contraction, secretion, neuronal ring and many other functions. Of particular interest are the oscillations seen in free intracellular calcium concentration, which are known to act as intracellular messages, relaying information within cells to regulate cell activity.
A key feature of intracellular calcium dynamics is that some physiological processes occur much faster than others. This leads to models with variables evolving on very di erent time scales. Using geometric singular perturbation techniques (GSPT) it is possible to exploit this separation in time scales to analyse the models. These techniques can be used to explain the observed dynamics, including oscillatory patterns known as mixed-mode oscillations and complicated bifurcation structures.
Chikungunya is a re-emerging mosquito-borne infectious disease that is spreading rapidly across Africa and Asia with new epidemics occurring in Europe and some Indian Ocean Islands. Two common mosquito species, Aedes aegypti and Aedes albopictus, which occur all over the world, are competent vectors for chikungunya virus. We design and analyze an ordinary differential equation model with mosquito dynamics for the spread of chikungunya. We parameterize the model using current literature, existing data, and lab experiments and compute the basic reproduction number. The spread of chikungunya is then compared to that of malaria. We show that malaria and chikungunya are sensitive to different parameters in the model, indicating that standard mitigation strategies for mosquito-borne diseases such as malaria may not work as well with chikungunya. We use sensitivity analysis to indicate where future research and mitigation efforts can focus for greatest effect in controlling the spread of chikungunya.
Understanding the geographic and temporal spread of food- borne diseases associated with fresh produce is crucial for informing adequate surveillance and control. As a first step towards this goal, we develop and analyze a three stage model at the processing/sanitization juncture in the fresh produce supply chain. The key feature of our model is its ability to describe basic dynamics of cross-contamination during wash procedures. We formulate general conditions under which our model predicts the potential for misdiagnosis of primary source contamination. We also discuss the importance of the model with regards to traceback studies, describing its ability to narrow parameter choices for detailed stochastic simulations as well as its "connect-ability" to models that include shipping and network dynamics. Finally, the model is useful for comparing various commercial biocidal wash procedures and is easily adaptable to include parameters such as temperature, turbidity, organic load, pH, etc.
Biological signals are often influenced by many physiological processes and functions. These generate different rhythms at different frequencies, such as cardiac cycles and respiratory oscillations in electrocardiographic recordings, and complex waveforms across a wide range of frequencies in electroencephalographic signals. These rhythms are typically not independent, and the cross-frequency coupling (CFC) between them can reflect important physiological interactions. CFC has been applied in many neurophysiological studies to detect physiological and pathological changes, and in particular phase-amplitude coupling has recently been used to illuminate the functionally specific coordination of neurophysiological activity on multiple scales. Traditional CFC methods usually assume linear and stationary signals that are composed of sinusoidal oscillations with constant frequency and amplitude. However, biological signals are frequently nonlinear and nonstationary, complicating the interpretation of CFC results. Furthermore, these methods usually require a priori specification of frequencies of interest, making them cumbersome for exploratory analyses of CFC. We have developed a new data-driven CFC analysis without any assumption of stationarity and nonlinearity. This method first identifies the rhythms present in a time series, and then quantifies the phase-amplitude modulation between rhythms at different frequencies. We have applied our method to simulated data and physiological signals including neural activity recordings of the circadian pacemaker (the suprachiasmatic nuclei) and electroencephalographic data. Compared to a traditional Fourier-based CFC analysis, this new method can better quantify nonstationary rhythms and their nonlinear interactions while avoiding the spurious detection of cross-frequency couplings that are an artifact of nonlinearities and nonstationarities.
Posters
Lake Erie is one of the Great Lakes in North America and has a favorable environment for agriculture in which nitrate (NO3) is widely used as fertilizers. On the other hand, it has witnessed recurrent summertime oxygen depletion and related microbial production of greenhouse gases such as nitrous oxide (N2O). In fact, N2O is an intermediate in denitrification, which is a microbial process of conversion of nitrate (NO3) to nitrogen gas (N2). This presentation will introduce the gene regulatory network and its (discrete) mathematical model of Pseudomonas aeruginosa, one of the microbes performing denitrification in Lake Erie. Polynomial Dynamical Systems (PDS) is used to model the network, and the model is analyzed by changing the concentration level of some environmental parameters such as oxygen (O2), nitrate (NO3) and phosphorus (P) to see how these parameters affect the long-run behavior of the network. Analysis is done in Analysis of Dynamic Algebraic Models (ADAM available at http://dvd.vbi.vt.edu/adam.html). Our goal is to generate some hypotheses leading us to the reason of accumulation of greenhouse gases in Lake Erie, which is still unknown.
- Seed dispersal and natural enemies both influence spatial patterns of seedlings, which in turn influence future abiotic and biotic interactions, with consequences for plant populations, distributions, and diversity. Clumped seed deposition is common, especially for vertebrate-dispersed seeds, and has the potential to significantly affect interactions with density-responsive enemies, yet has received relatively little attention.
- We used spatially explicit simulation models to examine how different patterns of seed dispersal and natural enemy attack structure seedling spatial patterns. We simulated clumped seed dispersal by combining a two-dimensional Student's T dispersal kernel for expected seed rain with a negative binomial distribution for seed deposition. We based our models for seed mortality on published data reflecting differing life histories of insect seed predators and soil-borne pathogens. We varied dispersal distance, degree of clumping, type of enemy, enemy dispersal distance, and fecundity among simulations.
- Under insect seed predation, seeds escaped predation by dispersing longer distances than insects, resulting in "Janzen-Connell" patterns in which seedling recruitment peaks at intermediate distances. When insects dispersed longer distances than seeds, higher seed densities near the tree satiated insects, resulting in "McCanny" patterns in which seed deposition, survivorship, and seedling establishment all decrease with distance from the parent tree. Total seedling establishment was lowest when insects and seeds dispersed similar distances.
- Under pathogen attack, Janzen-Connell patterns predominated except when seedling survival was virtually zero or one everywhere, or in the case where pathogen dispersal distances exceeded seed dispersal distances, producing "Hubbell" patterns in which seed deposition and seedling establishment decrease with distance, though survivorship increases.
- Clumped seed deposition increased the probability of seedling establishment under both insect seed predation and pathogen attack as it led to local satiation of insect seed predators and made it harder for pathogen distributions to track seeds.
- Synthesis. Our modelling study suggests that the relative dispersal distances of seeds and natural enemies are crucial to determining establishment rates and spatial patterns of seedlings. Better characterization of the movement and natural histories of natural enemies is critical to improving our understanding of seedling distributions and plant-enemy interactions.
Understanding how information propagates in a social network has been an active area of research over the last few decades. Early mathematical models were developed to understand how consensus is reached in simple networks. With the advent of social media which can connect millions of users, there has been resurgent interest in the problem. For instance, Frongillo et al. (arXiv:1109.5482v1, cs.SI, 2011), have examined how individuals in a social network can optimally exchange information about a dynamically changing parameter.
It is frequently assumed that discussion improves the decisions of individuals. However, such exchanges can correlate the beliefs of different individuals in a social network, and negatively impact a collective decision. Indeed, even individual decisions may suffer after an exchange of information. Bahrami group (Science Vol 329, pp.1081-5, 2010) observed that discussion can have a positive impact on decision making only if the observers have approximately equal competence. Redundancies are likely to be present in social networks, specifically in large networks with many edges. In such situations, how should observers integrate incoming information to achieve the best possible estimate and reach the best decisions?
We study a simple graphical Bayesian model of interacting observers who try to estimate a value in the outside world with their own observations and their local neighborhood information. We study in detail feed forward and recurrent network and establish a general result on how the individuals can achieve optimal Bayesian inference in the presence of redundancies. Such redundancies impact the performance of even optimal observers. We also consider the propagation of individuals biases and priors in making decisions and show how certain network structures impact an individuals' maximum- likelihood (ML) estimate.
Apart from our theoretical studies, we also design a lab-based psychophysical experiment to ask if humans are capable of optimally accounting visual sensory information in making decisions.
Pathogen- pathogen interaction, a form of epidemiological synergism, is an emerging arena of new research and understanding in studies of infectious disease in the health and clinical care. Pathogen interactions can be operational at different scales and is very common when they share a common host population. We focus on two closely related viruses in the Paramyxovirus family (data obtained from Department of Pediatrics and Biomedical Informatics, University of Utah):
- Respiratory Syncticial Virus (RSV), a cause of severe illness in very young children,
- Human Parainfluenza (HPIV), the current cause of the majority of cases of croup in the United States, which breaks into four distinct serotypes.
Though preliminary statistical analysis of correlation and regression on datasets indicates apparent interaction between them, but it does not exhibit the nature of interaction. To address the issue, we came up with two different hypotheses of interactions: cross-immunity and convalescence, and build up two different seasonally forced two-disease models. Using a variety of model-fitting approaches including trajectory-matching, probe-matching, and Bayesian methods, we estimate the strength of interactions along with other parameters such as amplitude of seasonality and rate of waning immunity.
In recent decades, we have seen an increasing reliance on mathematical models as complementary tools in dealing with outbreaks of infectious diseases ranging from the outbreaks of Malaria, AIDS/HIV, SARS and Influenza viruses, to name a few. As a result, inter- and trans-disciplinary scholars and public health officials have used theoretical models to assess the interplays between the dynamics of contact processes underlying the transmission mechanisms and the biological constraints of host-pathogen interactions. However, individual objective and subjective vulnerabilities to disease and their behavioural adaptations to infectious diseases have been ignored. Here, we formulate an in silico model of pathogen-avoidance mechanism, and investigate its impact on defensive behavioural measures (e.g., spontaneous social exclusions and distancing, crowd avoidance, voluntary vaccination adaptation, face mask wearing). Our analyses reveal complex relationships between spontaneous and uncoordinated behavioural changes, emergence of its contagion properties, and mitigation of infectious diseases. We find that the persistence and/or permanence of disease in the population can be impeded in the presence of effective behavioural changes. However, the required efficacy level for affecting endemicity of infectious disease is very high, ranging between 90-100%. Furthermore, it was found that under perfect effective behavioural change, there are three regions in the behavioural space where, (1) disease is always endemic even in the presence of behavioural change, (2) behaviour-Prevalence elasticity is observed and disease can sometime be eradication, and (3) eradication of endemic disease under permanence of habitual behavioural change is achieved. These results suggest that preventive behavioural changes (e.g., non-pharmaceutical prophylactic measures, social distancing and exclusion, crowd avoidance) are influenced by the salience of diseases and by individual differences in perception of risks. Additionally, these findings indicates that care needs to be taken when considering the effect of adaptive behavioural change in predicting the course of epidemics, and as well as the interpretation and development of the public health measures that account for spontaneous behavioural changes.
Tumor Angiogenesis, the process by which cancerous cells signal surrounding blood capillaries to form new capillaries and grow towards the tumor mass to vascularize it is necessary for the growth and spread of tumors. However, it possible to inhibit the process of signaling in order to prevent tumors from growing beyond a size that can be harmful.
Partial Differential Equations arising from this process can sometimes be stiff and thus care must be taken in using the appropriate numerical method to simulate.
In our work we explore some numerical methods that can be used to effectively solve such PDE's much faster. Our work is based on the model by H A Harrington et al. We however modified this model by including a source term to the PDE that describes the proliferation of Tumor Angiogenic Factors (TAFs).
We used three different Explicit Stabilized Runge-Kutta methods to solve the system numerically. We compared the CPU times of the various methods used on the model and also checked whether our results were consistent with the biological process.
Though sleep is a pervasive phenomenon within the animal kingdom, its function remains largely mysterious. The synaptic renormalization hypothesis is one leading theory which proposes that sleep is necessary to maintain homeostasis in the brain. According to this hypothesis, the overall coupling strength between neurons increases during waking, and one primary function of sleep is to reverse this process, or "renormalize" the brain's synapses. Experiments strongly indicate that such renormalization does occur, but exactly how the brain accomplishes this synaptic downscaling is unknown.
Here I propose a viable biophysical mechanism for synaptic renormalization. Simulations of large-scale neuronal networks show that modulation of a single chemical in the brain (acetylcholine) dramatically changes the dynamics of individual neurons by altering their phase response curves, which in turn leads to dramatic changes in the synchronization of large-scale neuronal networks. Through a spike-timing dependent plasticity rule, these changes in synchrony result in large differences in the overall strength of network connections, providing a possible biophysical mechanism for synaptic renormalization.
Leishmaniasis is a family of human infectious diseases spread by the bite of sand-flies. The Indian state of Bihar has the highest Leishmaniasis mortality rate in the world. Currently in Bihar, DDT (for controlling Leishmaniasis) is distributed (37.5 grams per individual, under WHO guidelines for achieving kala-azar elimination) only according to the size of the human population. Is the process of insecticide distribution optimal? In this research, we explore this question and present scenarios where insecticide allocation can be optimal using mathematical models. Since the species of the sand fly in India are zoophilic, we also considered the size of cattle population in the design of our models.
We use cost and epidemiological data, obtained from Bihar State Health Society through our collaborators at Rajendra Memorial Research Institute of Medical Sciences, Patna, Bihar, in our optimization model to identify the optimal amount of insecticide allocation. Our model treats insecticide spraying at the human as well as cattle dwellings (or sites). The analysis of the model recommends optimal site-dependent allocation of insecticide based upon a lowest cost-benefit ratio (where cost was related to insecticide material and benefit was increase in the sand fly death rate). We derive decision criteria and analytical relationships between socio-economic and epidemiological parameters and the optimal insecticide distribution.
This task of obtaining optimal allocation of resources is achieved systematically. Firstly, by minimizing only the cost of insecticide implementation and secondly, by minimizing both the cost and the number of Leishmaniasis cases. We hope the results from our analysis of various insecticide implementation scenarios might be helpful to the Bihar state public health department in designing effective Leishmaniasis control polices.
As a system of differential equations describing an epidemiological system becomes large with multiple connections between variables, the expressions for reproductive numbers become overly large and onerous to handle analytically. We present a new method which deconstructs the larger system into smaller subsystems, captures the bridges between the smaller systems as external forces, and bounds the full system reproductive numbers in terms of reproductive numbers of the smaller systems.
Contact tracing is an important method that has been used in the control of endemic contagious disease for decades, but we do not know much about the effectiveness or even the necessity of contact tracing yet. Contact tracing, followed by isolation, can reduce an additional number of infected individuals who are not removed by isolation process. That is the reason contact tracing is considered to be a useful method for reducing infections. Most of us believe that if some is good then more is better. So people intuitively deduce that high level contact tracing (those trace more contacts or last for longer period of time) results in having less infections than relatively low level contact tracing does. But that argument is not always the case according to our model.
We build a deterministic SIR model with isolation and contact tracing in the control of an epidemic disease. The model is applicable to diseases like smallpox, H1N1, SARS and some modern in?uenzas. We are able to conclude that tracing too many contacts during the contact tracing period might just postpone the outbreak of the disease and might NOT reduce infections effectively. Of course, it does not mean that we have the same kind of results for all infectious diseases and contact tracing strategies. There is a subtle connection between the appearance of the special case and the nature of the epidemic disease.
I will be interested in presenting some background of contact tracing, introducing the main model, demonstrating simulation results of different cases, explaining mathematically why the special cases appear and providing suggestions about how to choose a suitable contact tracing level. I'm also looking forward to knowing how well the model together with the conclusion and explanation can be understood by researchers from related areas, and any suggestions about either model improvement or future study will be appreciated very much.
It has been discovered that gene regulation is a strongly stochastic, yet still robust process. One well-known indicator for the robustness of gene regulatory networks is the so-called Derrida plot. For a fixed number of perturbations, the Derrida value is the expected Hamming distance after one time step. This value is generally small for networks that exhibit stable behavior and large for networks with more chaotic behavior. Thus far, attainment of these values has depended on time-consuming Monte Carlo simulations.
However, for networks based on multi-state nested canalizing functions - a class of functions found to be prevalent in gene regulation - explicit formulas for the Derrida values can be found; both for general multi-state nested canalizing functions and for Boolean nested canalizing functions of a particular Hamming weight.
Having actual formulas for Derrida values of networks governed by any nested canalizing function precludes simulation, which simplifies the use of Derrida plots for robustness investigations of complex networks. Thus, this research contributes to a better understanding of the robustness of gene regulation.
Johnes disease in cattle is caused by Mycobacterium avium subspecies paratuberculosis (MAP). MAP is associated with rapid weight loss and diarrhea of the infected cattle. This leads to reduced milk production in dairy farming and loses compounded by culling infected animals in order to prevent the disease from spreading within a herd. Disease progression follows four distinct stages, silent, subclinical, clinical and the advanced stage. Methods for early infection diagnosis are yet to be developed. The disease is hardly noticeable in the silent and subclinical stages. Therefore, the infection is noticed when animals are already in the clinical stage. This study investigates the "Iceberg phenomenon", which attempts to estimate the severity of the disease in a herd once an infected animal is identified. The phenomenon estimates that for one animal in the advanced stage, there are one-to-two in the clinical stage, four-to-eight in the subclinical stage and ten- to-fourteen in the silent stage. Using a system of differential equations our study shows that it is not possible to observe the Iceberg phenomenon using incubation periods associated with the natural course of disease progression. However, the phenomenon can be observed if the incubation period of the silent stage is assumed to be longer than that of the subclinical stage, and incubation period of the clinical stage is less than the subclinical incubation time. This gives rise to the question, is the biology of disease progression accurate or does the Icerberg phenomenon need to be revised?
The gypsy moth, Lymantria dispar (L.), is probably the most destructive forest defoliator in the North America. Gypsy moth outbreaks tend to be spatially synchronized over areas across hundreds of kilometers, which can greatly aggravate the ecological and socioeconomic impacts of high density populations and overwhelm management resources allocated to mitigate impacts. Outbreaks can result in loss of timber and other traditional forestry products. Greater losses tend to occur to the ecosystem services that forests provide, such as wildlife habitat, carbon sequestration, and nutrient cycling. Outbreaks can also change the composition of the community, including indirect changes to native herbivores that gypsy moths tend to outcompete and altering forest succession.
The United States can be divided in three different areas: A generally infested area (where gypsy moth populations are established), an uninfested area (populations are not established), or a transition zone between the two. There are different management programs matching these different areas: (1) Detection/ eradication, which targets new colonies in areas uninfested by the gypsy moth (e.g., the west coast of North America), (2) the Slow-the-Spread program, which consists of a barrier zone along the invasion front in the United States, and (3) suppression of outbreaks in areas that are infested by the gypsy moth as a means to mitigate impacts. This work focuses in optimal control techniques for models of areas where the population is established or in the invasion front.
We design an objective functional to minimize the cost generated by the defoliation caused by the population of gypsy moth and the cost of controlling the population with an aerial spray. The objective was to develop an optimal control framework and perform numerical simulations for various scenarios, that seeks to minimize the total cost due to gypsy moth (damage plus control cost).
Ecological network analysis (ENA), predicated on systems theory and Leontiev input-output analysis, is a method widely used in ecology to reveal holistic ecosystem properties. In the ENA framework ecosystems are modeled as weighted digraphs with tentacles, with the tentacles expressing the exchanges between the ecosystem's compartments and its environment. Several quantitative indicators have been formulated in ENA to evaluate ecosystems' functions and health. Network particle tracking builds on ENA to o er a Lagrangian particle method that describes the activity of the ecosystem at the microscopic level. NPT provides access to the pathway data of individual particles that ow in the ecosystem; thus opens the door to using Lagrangian methods in ENA. We use NPT to compute two ecosystem's indicators (cycling and through ow) and validate our ndings on actual ecosystems data. While the traditional ENA cycling index and through ow matrix are computed under steady-states assumptions, the Lagrangian methodology a orded by NPT paves the way for the development of dynamic ecosystem's indicators.
Calcium waves, triggered by input to metabotropic glutamate receptors, spread slowly throughout a portion of a neuron's dendritic tree by calcium-induced calcium release (CICR) from intracellular stores in the endoplasmic reticulum (ER). As these waves spread, they modulate ion channel activity, altering dendritic integration. These chemical dynamics are themselves modulated by electrical signaling, as back-propagating action potentials raise intracellular calcium which primes the stores.
To study this interaction and other questions from the intersection of neuroscience and cell biology, we extended the NEURON simulator's support for reaction-diffusion models. We added support for the systems biology markup language (SBML) to work with models from the cell biology community, for stochastic diffusion for the low molecule counts common in spines, and for three-dimensional simulations to study the role of geometry.
We then considered a model with the combined ER and plasma membrane channel dynamics which demonstrates facilitation of wave initiation due to priming via back-propagating action potentials. The ionic mechanisms responsible for CICR waves are disrupted in certain pathological conditions including Alzheimer's Disease, and we discuss the model's predictions of how these changes affect electrical signaling.
Many recent studies have shown that the initiation of human cancer is due to the malfunction of some genes at the R-checkpoint during the G1-to-S transition of the cell cycle. Identifying and modeling the dynamics of these genes has a paramount advantage in controlling and, possibly, treating human cancer. In this study, a new mathematical model for the dynamics of a cancer sub-network concentration is developed. Positive equilibrium points are determined and rigorously analyzed. We have found a condition for the existence of the positive equilibrium points from the activation, inhibition and degradation parameter values of the dynamical system. Numerical simulations have also been carried out. These results confirm analyses in the literature.
Malaria remains one of the biggest killers in developing countries. It is caused by Plasmodium parasites and transmitted from human to human by the female Anopheles mosquito. Mosquito control provides one effective way of containing malaria. We apply optimal control theory to a deterministic ordinary differential equation model for the dynamics of malaria transmission in which partially immune humans also contribute in disease transmission. Different control measures, including insecticide-treated bed-nets, indoor residual spraying, and treatment are investigated, and the most effective control measure, which can be implemented at the least possible cost, is identified. The impacts of insecticide-treated bed-nets and indoor residual spraying on disease transmission and mosquito mortality are explored.
Necrotizing enterocolitis is an intestinal inflammatory disease that is a major cause of death in premature infants. A recently developed mathematical model of cell layer migration during experimental necrotizing enterocolitis based on an assumption of elastic deformation of the cell layer leads to a generalized Stefan problem. Analysis and numerical results indicate that a large class of constitutive equations for the dependence of proliferation on stretch leads to traveling wave solutions with constant wave speed.
HIV/AIDS disease continues to spread alarmingly despite the huge amounts of resources invested in fighting it. There is a need to integrate the series of control measures available to ensure a consistent reduction in the incidence of the disease pending the discovery of its cure.We present a deterministic model for controlling the spread of the disease using change in sexual habits and antiretroviral (ARV) therapy as control measures. We formulate a fixed time optimal control problem subject to the model dynamics with the goal of finding the optimal combination of the two control measures that will minimize the cost of the control efforts as well as the incidence of the disease.We estimate the model state initial conditions and parameter values from the demographic and HIV/AIDS data of South Africa. We use Pontryagin's maximum principle to derive the optimality system and solve the system numerically. Compared with the practice in most resource-limited settings where ARV treatment is given only to patients with fullblown AIDS, our simulation results suggest that starting the treatment as soon as the patients progress to the pre-AIDS stage of the disease coupled with appreciable change in the susceptible individuals' sexual habits reduces both the incidence and prevalence of the disease faster. In fact, the results predict that the implementation of the proposed strategy would drive new cases of the disease towards eradication in 10 years.
Cyclic patterns of neuronal activity are ubiquitous in neural systems of almost all animal species. To elucidate the underlying dynamical mechanisms for the storage and retrieval of cyclic patterns in neural networks is fundamentally important for understanding the origin of rhythmic movements such as locomotion, respiration, swallowing, and scratching etc. In this talk, we summarize our investigations in the storage and retrieval of binary cyclic patterns in continuous, asymmetric Hop eld-type networks using the pseudoinverse learning rule. Admissibility conditions that guarantee a connection matrix satisfying the transition conditions for the cyclic patterns are formulated and proved. Based on the structural analysis of the vector space spanned by the row vectors of an admissible cycle, admissible cycles are classi ed into three types: Simple cycles, separable, and inseparable composite cycles. Networks constructed by composite cycles can be partitioned into disjoint clusters, and each cluster corresponds to a simple cycle component. If the networks are constructed by separable cycles, clusters in these networks are completely isolated. Consequently, the dynamics of such networks is determined by that of the isolated clusters. A Hopf bifurcation analysis shows that an attracting limit cycle, which may correspond to the prescribed cycle, arises in every network constructed by a minimal simple cycle with row vectors containing the same number of 1's and 1's. In networks with ring topology, the complete description of bifurcations is derived and it is shown that the transitions from xed points to stable/unstable limit cycles are shown to be multiple saddle-nodes on limit cycle bifurcations. Implications of our theoretical ndings for rhythmogenic mechanism will be brie y discussed as well.
We study the biophysical and mathematical mechanisms underlying the metachronal rhythm exhibited in crayfish's forward swimming: rhythmic movements of the pleopods on the crayfish abdomen progress from back to front with the same period, but neighboring pleopods are phaselagged by 25% of the period. This coordination of limb movements is maintained over a wide range of frequency.
The underlying neural circuit is a chain of half-center oscillators (HCOs), which are endogenous local rhythm generators that produce periodic outputs controlling the motion of pleopods. Within each HCO are two non-spiking neurons coupled by mutual inhibition. Using a Morris-Lecar-type model, we first study the phase response property of each HCO and show that different local rhythmogenic mechanisms give rise to strikingly different local phase response properties. Using weakly coupled oscillator theory, we then study how the metachronal rhythm is produced and provide an explanation on why only certain combinations of network connectivity and local rhythmogenic mechanisms are able to produce the observed 25% phase-locking metachronal rhythm. Our results are obtained under the assumptions that each HCO is a relaxation-like oscillator and that the synaptic connections are instantaneous and have a sharp threshold of activation.
