September 9 - 13, 2013
What is autoregulation?
Autoregulation is a biological process in which an internal adaptive mechanism works to adjust (or miti-gate) an animal’s response to stimuli. For example, the autoregulation process results in the maintenance of blood flow to tissues at a certain level despite variations in blood pressure or metabolism. Autoregulation is most prominent in the kidney, the heart, and the brain, inasmuch as appropriate perfusion of these organs is essential for life, and through autoregulation the body can divert blood (and thus, oxygen) where it is most needed.Why study the kidney?
Autoregulation of renal blood flow and glomerular fil-tration rate in the kidney is critical, since about 25% Pressure of one’s cardiac output passes through the kidney. The kidney is responsible for excreting a small (but appro-
priate) fraction of the filtrate depending on the condi-tions of the body. Impaired renal autoregulation is a symptom of and a contributing factor to the progress of diseases such as hypertension and diabetes. The frequencies of these diseases have skyrocketed among the US and overseas population in recent decades. By gaining a better understanding of renal autoregulation, in physiological and pathophysiological conditions, we might have a better idea of how to control the progres-sion of hypertension and diabetes.What are the autoregulation mechanisms in the kidney?
The two major autoregulation mechanisms in the kidney are the tubuloglomerular feedback (TGF) and myo-genic response. TGF attempts to balance glomerular filtration with tubular reabsorptive capacity, whereas the myogenic mechanism induces vasoconstriction when blood pressure is increased. These two mechanisms
share a common effector in the afferent arteriole, which is the vessel that delivers blood to the kidney’s filter (glomerulus).What will this team do?
We will develop a model that represents the TGF and myogenic mechanisms. The model will include mass transport along a loop of Henle (PDE) and a circuit-based representation of the afferent arteriole (algebraic equations). There are a lot of interesting things we can do with the model:
What do you need to know to be part of this project?
- Bifurcation analysis. TGF is a negative feedback system with a nonzero feedback delay. One can ask the question: when the system is given a transient perturbation, will it return to a steady state, or will it evolve into a sustained oscillation (i.e., limit-cycle oscillation)? How does the answer depend on model parameters (i.e., feedback gain and delay, myogenic response strength, etc.)? One can study this question by analyzing a linearized version of the model equations.
- Numerical simulations. The full nonlinear model can be run, for different sets of parameters, to validate predictions of the linear model above.
- Stochastic PDE with feedback. We can introduce stochasticity into some of the model parameters and see how the stability of the system is affected.
- We can use the model to study pathophysiological mechanisms in diabetes, including salt-handling and glomerular hyperfiltration.
You should be interested in the physiology, but you do not need to be an expert on the kidney. You should know how to code in matlab.
During sleep, we experience a loss of consciousness coupled with decreased muscle tone and increased sensory thresholds. Yet sleep is not a uniform state. For most mammals, sleep involves alternations between the rapid eye movement (REM) sleep in which we have our most vivid dreams and non-rapid eye movement (NREM) sleep including deep slow wave sleep.
These two sleep states are very different physiologically. NREM sleep is a state of reduced brain activity, consistent with notions of sleep as rest, and it has been found that slow-wave activity during NREM sleep is homeostatically regulated with respect to the duration of the preceding wake episode. During REM sleep on the other hand, brain metabolism is as high as during wake or even higher. REM is distinguished from both NREM and wake by an absence of muscle tone and a suspension of thermoregulation.
The ultradian rhythm of REM/NREM cycling is often described as a 90-minute sleep cycle in humans, but actually the data is much more variable than that description suggests. In fact, many mathematical models of REM/NREM dynamics in mammals have been purely stochastic [1,5,6].
These stochastic models capture many features of the data but are phenomenological, not taking into account the biological substrate. Recent mathematical models focusing on the biological mechanisms underlying REM/NREM cycling have been deterministic [3,11].
This project aims to develop a model for REM/NREM dynamics within sleep/wake cycling that carefully combines deterministic and stochastic elements. We will build on recent neurophysiological models (such as [7,10,12]) by considering how the stochastic dynamics may arise naturally in the biological mechanisms. We will test the model against data used for stochastic models, such as the statistics of state transitions and durations.
We will then use the model to explore some current hypotheses concerning the regulation of REM sleep such as that REM sleep propensity increases primarily during NREM sleep , that REM sleep propensity increases during both wakefulness and NREM sleep , that REM cycling drives the REM/NREM cycles , and that thermoregulation and energy management determine REM bout durations .
While the functions of sleep remain mysterious, progress is being made in understanding the mechanisms by which sleep and its substates are regulated. This project aims to contribute to that understanding with respect to REM/NREM cycling, providing insights that bring us closer to identifying the purpose of REM sleep.
This project will use methods from dynamical systems and stochastic processes, some basic statistics, and numerical simulation.
 Bassi A, Vivaldi E, Ocampo-Garces A. The time course of the probability of transition into and out of REM sleep. Sleep 32(5):655-669, 2009.
 Benington JH. Debating how REM sleep is regulated (and by what). J Sleep Res 11:29-33, 2002.
 Diniz Behn C, Ananthasubramaniam A, Booth V. Contrasting existence and robustness of REM/non-REM cycling in physiologically based models of REM sleep regulatory networks. SIAM J App Dyn Sys 12(1):279-314, 2013.
 Franken P. Long-term vs. short-term processes regulating REM sleep. J Sleep Res 11:17-28, 2002.
 Gregory G and Cabeza R. A two-state stochastic model of REM sleep architecture in the rat. J Neurophysiol 88:2589-2597, 2002.
 Kim JW, Lee J-S, Robinson PA, Jeong D-U. Markov analysis of sleep dynamics. Physical Review Letters 102:178104, 2009.
 Kumar R, Bose A, Mallick BN. Mathematical model towards understanding the mechanism of neuronal regulation of wake-NREMS-REMS states. PLOS One 7(8):e42059, 2012.
 Kumar VM. Body temperature and sleep: are they controlled by the same mechanism? Sleep and Biol Rhythms 2:103-124, 2004.
 LeBon O. Which theories on sleep ultradian cycling are favored by the positive links found between the number of cycles and REMS? Biol Rhythm Res DOI:10.1080/09291016.2012.721590.
 Phillips AJK, Robinson PA. A quantitative model of sleep-wake dynamics based on the physiology of the brainstem ascending arousal system. J Biol Rhythms 22(2):167-179, 2007.
 Phillips AJK, Robinson PA, Klerman EB. Arousal state feedback as a potential physiological generator of the ultradian REM/NREM sleep cycle. J Theoretical Biol 319:75-87, 2013.
 Rempe M, Best J, Terman D. A mathematical model of the sleep/wake cycle. J Math Biol 60:615-644, 2010.
Clustering is a stable phase-locked activity pattern that has been observed in networks of intrinsically oscillating neurons. In these states the network breaks up into clusters. Neurons within a cluster exhibit phase-locked behavior with zero phase-lag, while between clusters neurons are phase-locked with non-zero phase-lag. Clustering has been proposed as a way of inducing network level oscillation frequencies which individual oscillators are incapable of producing (Kilpatrick & Ermentrout, 2011). In networks with all-to-all coupling, multiple stable cluster states can co-exist (Golomb & Rinzel, 1994). This multistability enables the network the flexibility to respond to different inputs with different frequencies.
The existence and stability of cluster states has been studied using phase model reductions (Okuda, 1993), direct analysis of the network (Choe et al., 2010), numerical simulation and numerical bifurcation analysis (Golomb & Rinzel, 1994). The majority of this work has been restricted to networks of identical neurons with all-to-all coupling. Indeed, the existence of clustered states can be attributed to the inherent symmetry in such systems (Ashwin et al., 2010). However, clustered states have also been observed in more realistic models. For instance, Li (2003) showed that heterogeneity in the coupling strengths can stabilize otherwise unstable phase-locked states. Chadwick (2005) showed that clustered states which are stable in an all-to-all coupled network of identical neurons can persist in the presence of mild heterogeneity of the oscillator frequencies and the loss of some connections in the network.
A more systematic study of clustered states in more realistic networks is needed. This project will start to fill this gap by quantifying how various factors affect the existence and stability of cluster states, the corresponding network frequencies, and the presence of multistability. The exact factors considered will depend on the interests and background of the participants. Potential factors are: the presence of heterogeneity (e.g. in the firing frequencies of the individual neurons), network properties (topology, type of coupling, coupling delay), intrinsic cell properties (type I vs type II oscillators, presence of spike frequency adaptation) and presence of noise.
This project will include opportunities for those with interests in any or all of the following: mathematical analysis (stability, bifurcations, perturbation theory, network theory), numerical simulation and numerical continuation both for deterministic systems and systems with noise.
P. Ashwin, G. Orosz & J. Borresen (2010) Chapter 3 of Nonlinear Dynamics and Chaos: Advances and Perspectives.
J. Chadwick (2005) Summer Research Report. University of Waterloo.
C.-U. Choe, T. Dahms, P. Hoevel & E. Schoell. (2010) Physical Review E 81 025205.
D. Golomb & J. Rinzel (1994) Physica D 72(3): 259-282.
Z.P. Kilpatrick and B. Ermentrout (2011) PLoS Computational Biology 7(11) e1002281.
Y.-X. Li, Y.-Q. Wang & R. Miura (2003) J. Computational Neuroscience 14(2): 139-159.
K. Okuda (1993) Physica D 63: 424-436.
Anti-coagulation therapy is often prescribed after surgery in order to prevent
the formation of dangerous clots that could cause strokes or respiratory obstructions. However, the clotting properties of blood are naturally tightly regulated by the healthy body, and these treatments can upset these innate regulatory processes. Furthermore, anti-coagulants have negative side effects such as excessive bleeding. For example, people taking anti-coagulants can experience dangerous clotting after airplane flights, despite a lack of clinical evidence that flying adversely affects clotting activity. In this project, we hope to build mathematical models to address several key questions about the administration of these anti-coagulant drugs. Some of these questions are:
- What should the dosing be: how much drug should be given, and how
- How do current monitoring practices reflect actual anti-coagulation activity (for example, how good of a proxy is the INR)?
- How do we model pressure changes that reflect airplane flight (ground
level air pressure versus the pressure in an airplane that is flying at 30,000
- How would blood vessel damage affect clotting? Can scar tissue in the
blood vessels or damage to the vascular pumps have a significant effect?
Currently, mathematical models of blood clotting exist. As a starting point, we
propose to add a PK-PD model of a common anti-cogulant, such as Warfarin or Coumadin, to exiting models of blood clotting, such as the spatial model of
Choose a common anticoagulation therapy (such as Warfarin or Coumadin). Add the associated PK/PD representing Coumadin action to the Fogelson-Leiderman spatial blood clot formation model. Heuristic optimization procedures could be used to answer the doing question. Simulations could illuminate the relationship between commonly used proxies for anit-coagulation activity and what is actually happening. Finally, model parameters can be adjusted to represent changes in the host’s environment, such as the change in pressure experienced during an airplane flight, or the changes in tissue properties that follow major surgery or trauma.References
 A. J. M. dePont, J. H. Hofstra, D. R. Pik., J. C. M. Meijers and M. J. Schultz. Pharmacokinetics and pharmacodynamics of danaparoid during continuous venovenous hemofiltration: a pilot study. Critical Care 11:R102 (2007). (doi:10.1186/cc6119)
 K. Leiderman and A. Fogelson. Grow with the flow: a spatialtemporal model of platelet deposition and blood coagulation under flow. Mathematical Medicine and Biology 28: 47?84 (2011) (doi:10.1093/imammb/dqq005).
 E. Pasterkamp, C. J. Kruithof, F. J. M. Van der Meer, F. R. Rosendaal and J. P. M. Vanderschoot. A model-based algorithm for the monitoring of long-term anticoagulation therapy. Journal of Thrombosis and Haemostasis, 3:915921 (2005).
 S. Thijssen, A. Kruse, J. Raimann, V. Bhalan, N. W. Levin and P. Kotanko
. A Mathematical Model of Regional Citrate Anticoagulation in Hemodial-
ysis. Blood Purif. 29:197203 (2009). (doi: 10.1159/000245647)
 R. Vink, R. A. Kraaijenhagen, M. Lei and H. R. Büller. Individualized
duration of oral anticoagulant therapy for deep vein thrombosis based on a decision model. Journal of Thrombosis and Haemostasis, 1: 2423–2530 (2003)
 F. F. Weller. A free boundary problem modeling thrombus growth. Model
development and numerical simulation using the level set method. J. Math.
Biol., 61: 805–818 (2010) (doi:10.1007/s00285-009-0324-1).Project Leaders
Ami Radunskaya, (Pomona College, Claremont CA 91711)
Lisette de Pillis (Harvey Mudd College, Claremont CA 91711)
Erica Graham (North Carolina State, Raleigh NC )
Recent developments in microfluidic devices have enabled controlled studies and manipulation of fluid flows with length scales at the micron level . At this length scale, viscous forces are very important and processes such as diffusion and surface tension dominate. In many microfluidic experiments that measure chemical and biological processes, mixing of the fluid within the chamber is desirable. Turbulent mixing and pumping do not work in this microscale world where inertia is negligible. For example, two laminar streams flowing in contact with each other will not mix except by diffusion. Strategies for generating mixing at the microscale are therefore an important component of experiment design. A novel approach to microfluidic mixing introduced by N. Darnton et al  is the use of flagellated bacteria as fluidic actuators. In this approach, large numbers of bacteria are made to adhere to a substrate. The adherent bacteria on this “bacterial carpet” freely rotate their flagella, which move the fluid near them and act as microscopic propellers. Furthermore,
they can live on small amounts of simple nutrients (e.g., sugars) and can even maintain mobility for several hours without food . Since no external power source is needed, the use of bacteria can be advantageous over conventional micro- or nano-fabricated devices.
The objective of this project is to study the flow induced by the collective flagellar motion of bacterial carpets. In particular, we will model a doubly-periodic array of rotating helical flagella stuck to a surface immersed in a viscous, incompressible fluid. We will examine the transport of fluid particles above the carpet as a function of flagellar distribution and geometries. In addition, since many applications in microfluidics involve suspended macromolecules and biopolymers, we will examine the transport of particles of non-zero volume as well as fibers. Our mathematical model and computational framework will be based upon the method of regularized Stokeslets [1, 2], which
was designed to study the coupling of elastic structures with fluid at zero Reynolds number, and a more recent extension of this method to incorporate periodicity .
 R Cortez. The method of regularized stokeslets. SIAM J. Sci. Comput., 23:1204, 2001.
 R Cortez, L Fauci, and A Medovikov. The method of regularized stokeslets in three dimensions:
analysis, validation, and application to helical swimming. Phys. Fluids, 17:031504, 2005.
 N Darnton, L Turner, K Breuer, and HC Berg. Moving fluid with bacterial carpets. Biophys.
J., 86:1863–1870, 2004.
 M Kim and K Breuer. Use of bacterial carpets to enhance mixing in microfluidic systems. J.
Fluids Engr., 129:319, 2007.
 K Leiderman, EL Bouzarth, R Cortez, and AT Layton. A regularization method for the nu-
merical solution of periodic stokes flow. J. Comput. Phys., 236:187–202, 2013.
 HA Stone, AD Stroock, and A Ajdari. Engineering flows in small devices: Microfluidics toward
a lab-on-a-chip. Annu. Rev. Fluid Mech., 36:381–411, 2004.
1. Motivation of the Problem
Intermittent preventive treatment (IPT) describes the process in which a full therapeutic course of anti-malarial drug is administered to individuals most vulnerable to malaria effects, regardless of their infection state. IPT usage is increasingly being utilized as a means of preventing malaria in these at risk humans in malaria endemic regions. The vulnerable populations are usually defined by age in which case if IPT is administered to infants it is referred to as (IPTi), to children (IPTc) and to pregnant women (IPTp). In fact, IPTp is currently the recommended preventative approach for malaria in pregnant women, and is being explored as a way to prevent malaria in infants and children.
Mathematical models for malaria that include IPT can be used to predict rapid spread for the malaria parasite with IPT use. Our goal is to investigate the relationship between IPT and the spread of drug resistance.2. Objectives
- Develop an age structured model to investigate the relationship between IPT and spread of drug resistance to malaria.
- Determine the critical level of IPT treatment that would minimize the spread of drug resistance and reduce disease prevalence and burden.
- Determine the optimal timing for IPT drug administration.
- Identify the targeted age groups for optimal treatment strategies.
Tools from dynamical systems theory will be used to analyze the properties of the model, and the results can help address the biological questions proposed in this project. Numerical simulations of the model will also be conducted to confirm or extend the analytical results. One of the objectives will be for the group to continue to work on the project after the workshop and to develop it into a manuscript for publication.3. Suggested Readings (will continually be updated)
O'Meara W.P., Smith D.L. and McKenzie F.E. (2006). Potential impact of intermittent preventive treatment (IPT) on spread of drug resistant malaria. PLoS Medicine 3(5): 0633-0642.
Carneiro I. Smith L. Ross A. et. al. (2010). Intermittent preventive treatment for malaria in infants: a decision-support tool for sub-Saharan Africa. Bull World Health Organ 88:807-814.
Koella J.C. and Antia R. (2003). Epidemiological models for the spread of anti-malarial resistance. Malar J. 2:3.
Bloland P. (2001). Drug resistance in malaria [WHO/CDS/CSR/DRS/2001.4], Geneva: World Health Organization, 2001.
Hastings I.M. (1997). A model for the origins and spread of drug-resistant malaria. Parasitology 115:133-141.
Hastings I.M., Watkins W.M. and White N.J. (2002) The evolution of drug-resistant malaria: The role of drug elimination half-life. Philos Trans R Soc Lond B Biol Sci 357: 505-519.
Hastings I.M. and Watkins W.M. (2005). Intensity of malaria transmission and the evolution of drug resistance. Acta Trop 94: 218-229.
Evolutionary diversification is one of the central concepts in ecology. The mechanisms leading to diversification of species in geographic isolation are well understood. There is however another scenario of diversification (sympatric diversification) whereby speciation occurs without the geographical separation of diverging populations. While it is becoming increasingly apparent that sympatric diversification is an important source of biological diversity, its underlying mechanisms are poorly characterized.
In two recent papers [1,2], experimental results are reported where sympatric diversification was observed in several E. coli bacterial systems. The observed evolutionary dynamics were driven at least in part by a co-‐evolutionary process, in which mutations causing one type of physiology changed the ecological environment, which in turn allowed the invasion of mutations causing an alternate physiology. The parallel genetic changes underlying similar phenotypes in independently evolved lineages provided the first ever empirical evidence of adaptive diversification as a predictable evolutionary process 
In this project, we will work on creating a mathematical model that can describe the observed data. In particular, we will work on an explanation of the observed coexistence of two different types of bacteria in the same spatial location. As a reference point, we will use existing mathematical models of adaptive diversification due to frequency-‐dependent ecological interactions.
The project will involve the parts: (1) Exploration/model construction, (2) Model validation by comparing with the experimental data in [1,2], (3) Explanation of the observed process of diversification and coexistence. The work will include both analytical and numerical components.
 Herron, Matthew D., and Michael Doebeli. "Parallel evolutionary dynamics of adaptive diversification in Escherichia coli." PLoS biology 11.2 (2013): e1001490.
 Burgess, Darren J. "Evolution: Experimental evolution probes neighbourly niches." Nature Reviews Genetics (2013).
Practically all chemotherapeutic agents and potentially many targeted therapies that are used in the clinical treatment of cancer lead to drug resistance. There is, however, no consensus on whether drug resistance is pre-existing or acquired. Pre-existing drug resistance means the cancer contains a subpopulation of drug resistant cells at the initiation of treatment, and that these cells become activated or selected for during the course of therapy. On the other hand, acquired resistance involves the tumor gradually developing drug resistance due to drug action and other factors, such as microenvironmental or metabolic conditions.
Mechanisms of drug resistance are currently being studied in cell culture. Biologists produce drug-resistant cell lines by exposing the cells to the drug, collecting the surviving cell subpopulation, and repeating this process through several passages until the remaining subpopulation of cells no longer responds to the treatment. While this is an effective way to generate a resistant cell population, this in vitro process does not reveal whether the surviving cells become more resistant to the chemotherapeutic treatment with each cell passage, or if a small population of susceptible cells was present from the beginning and
simply has overgrown the other cells during the course of the experiment.
This poses several questions. If the hypothesis of a pre-existing population of resistant cells is true, what mechanisms enable those cells to resist the drug action of the often multiple chemotherapeutic treatments that may be given to a patient sequentially or in parallel? If the hypothesis of gradual emergence of drug resistance is true, what factors contribute to the development of acquired drug resistance?
(a) Develop a hybrid discrete-continuous model of cancer response to a single drug or drug combinations based on tumor histology and immunohistology.
(b) Simulate and compare outcomes of possible resistance mechanisms, such as pre-existing populations of resistant cells; role of cancer stem cells; drug penetration of tumor tissue architecture; dynamics of drug absorption and efflux; role of irregular metabolite gradients.
(c) Determine a hierarchy of factors that contribute to tumor resistance to anti-cancer drugs.
We will use the agent-based techniques (such as cellular automata or particle-spring models) and partial differential equations to develop a hybrid model of tumor response to chemotherapy. We will employ image analysis and classification methods to analyze histology images and define model initial conditions. Computational simulations will be conducted to investigate model outcomes under various drug resistance mechanisms.
Tredan O, Galmarini CM, Patel K, Tannpck IF, Drug resistance and the solid tumor
microenvironment, J. Natl Cancer Inst, 2007, 99:1441-1454
Dean M, Fojo T, Bates S, Tumor stem cells and drug resistance, Nature Reviews Cancer, 2005, 5:275-284
Lambert G, Estevez-Salmeron L, Oh S, Liao D, Emerson BM, Tlsty TD, Austin RH, An
analogy between the evolution of drug resistance in bacterial communities and malignant tissues, Nature Reviews Cancer, 2011, 11:375-382
Phototransduction cascades transform light to electrical signals that project to the brain through a sequence of chemical reactions. Image forming vertebrate vision is facilitated by rods and cones in the back of the retina, each of which has its own opsin, a light sensitive G-protein coupled receptor. Recently a third opsin, melanopsin, was identified, not in rods and cones, but in intrinsically photosensitive retinal ganglion cells (ipRGCs). It is thought that melanopsin is primarily responsible for non-‐image forming visual functions such as the regulation of sleep cycles and pupillary light reflex.
Since melanopsin was only discovered recently, not much is known about its phototransduction pathway, which involves both an activation and inactivation pathway, however, it is believed that melanopsin uses a pathway similar to those found in invertebrate retinas. Data suggests that the melanopsin pathway consists of an activated G-‐protein, which in turn activates a phospholipase‐C (PLC). PLC breaks down PIP2 into second messengers, which leads to an increase in the intracellular calcium, and in turn depolarizes the cell. Inactivation of the melanopsin is hypothesized to be a two part process that first involves phosphorylation of its carboxy tail and second binding by a beta-‐arrestin because melanopsin is a G-‐protein coupled receptor.
Based on modeling of rhodopsin, a well-known and well-‐studied rod photopigment, a differential equation based model of the activation and inactivation of melanopsin has been formulated and tested. Both the activation and the inactivation can be described by a phototransduction cascade, which consists of a series of chemical reactions. For both activation and inactivation, the model has been tested against two types of experimental data: electrophysiology data and calcium imaging data from melanopsin cultivated in human embryonic kidney (HEK) cells. Model parameters that weren’t obtained from experimental data were fit using the wild type data. The model of inactivation, with the parameters from the wild type data, accurately predicted experimental responses to an over expression of beta arrestin. Current work involves coupling the two models to predict the response to a second flash of light that experimentally produces a quicker response than the first flash, yet with lower amplitude, a process known as adaptation.
Hamer et al (2003) developed a model for rhodopsin phototransduction cascade that included stochastic mechanisms in which the probability that a molecular species engages in a particular reaction is proportional to the reaction rate and then use Monte-Carlo simulations and the Gillespie method to accurately predict activation and inactivation of rhodopsin. In this project, I propose to extend the methods suggested by Hamer et al for rhodopsin to melanopsin.
Hamer, Nicholas, Tranchina, Leiderman, Lamb, Multiple Steps of Phosphorylation of Activated Rhodopsin can account for the Reproducibility of Vertebrate Rod Single-photon Responses, J. Gen. Physiology, vol 122, 419‐444, 2003.