Mathematical Biology


Growth cone image
Endothelial cell image

Xenopus oocytes
Growth cones

Image by John Jellies at the Department of  Biological Sciences,
Western Michigan University
Cytoskeletons of human endothelial cells glow green in this immunofluorescent micrograph 
Image by Sui Huang and Donald E. Ingber, Harvard Medical School.
Xenopus oocytes

Image by Erwin Sigel, University of Bern


My research interest is in Mathematical Biology, particularly mathematical modeling in  Cell Biology and Physiology.   This page  contains the following information: list of publications, a description of  research interests (calcium signaling, bacterial pattern formation, endothelial cell deformation, axon guidance), my graduate training in mathematical biology and research presentations created with Ppower4.


Publications


"The effect of residual Ca2+ on the stochastic gating on Ca2+-regulated Ca2+ channel models", B.~Mazzag, C.~Tignanelli and G.~D.~Smith, Journal of Theoretical Biology, 235:121-150, 2005. (pdf.)
                                                                                              
"Model of Bacterial Band Formation in Aerotaxis", B.Mazzag, I.Zhulin, A.Mogilner, Biophys.J.,2003, 85(6):3558-3574. (.pdf)

"A Model for Shear Stress Sensing and Transmission in Vascular Endothelial Cells", B.Mazzag, J.S.Tamaresis, A.I.Barakat,
Biophys.J., 2003, 84(6):4087-4101. (.pdf)

"Mathematical analysis of the swarming behavior of myxobacteria", A.Gallegos, B.Mazzag,A.Mogilner, accepted for publication at the Bulletin of Math Biology. (.pdf)


Calcium signaling

My postdoctoral work  with Greg Smith on calcium signaling has focused on so called "Ca2+-regulated Ca2+channels" (such as IP3 receptors or ryanodine receptors) that release Ca2+ into the cell body from an internal store, and whose opening and closing is also regulated by the Ca2+ concentration  near the channel mouth. We are studying the effect of the localized Ca2+ domain that develops near the channel mouth on the open probability of the channels.   First, we developed a model for a single, stochastically opening channel and investigated, using Monte Carlo simulations, how its open probability changed.  In addition, we obtained
analytical estimates of the open probability of the channel in the limit that the domain is very fast or very slow compared to the channel kinetics. Next, we extended this work to a cluster of channels where each channel is a stochastically active point source that contributes to the Ca2+-domain described by a 3D reaction-diffusion equation. 



PuffSimulation


The above figure shows results related to this work. It is a snapshot in time of the numerical simulation in which 19 channels contribute to the localized Ca2+ domain by opening and releasing Ca2+. The top 3 panels show the number channels in closed (C), open (O) or in refractory/inactivated (R) state as a function of time. The vertical line in these figures indicates the current position in time.  The bottom left panel depicts the spatial position of the channels and their state at the current time (approximately 46 ms).  The bottom left panel shows the [Ca2+] at this time at a given distance above the channels.

Graduate Training in Mathematical Biology

During my graduate career at the University of California, Davis I participated in the Research Training Grant for Nonlinear Phenomena in Biology (RTG). In addition to my mathematics course work and research with my advisor, Alex Mogilner, I also completed RTG core courses and requirements.  These courses have had a great influence on my development as a researcher and as a teacher.
                                                                                               
As part of the RTG program, I completed a lab rotation on mechanical deformation of endothelial cells with Abdul Barakat in the Mechanical and Aeronautical Department at UC Davis.  (He is also affiliated with Biomedical Engineering at UC Davis.) To satisfy another RTG requirement, I participated in a student group project called Second Year Project.  Work on this project lead me to an internship with Geoff Goodhill at the Neuroscience Department of Georgetown University.  Our work on mathematical modeling of axon guidance also contributed to my dissertation.  Due to my exposure to a number of interesting biological problems, my dissertation work is a collection of three projects that are tied together by the mathematical ideas. 


 Bacterial Pattern Formation I.
Aerotaxis

 My first research project with my advisor was on mathematical modeling of bacterial aerotaxis.  Aerotaxis is the particular form of chemotaxis in which oxygen plays the role of both the attractant and the repellent.  Aerotaxis occurs without methylation adaptation, and it leads to fast and complete aggregation toward the most favorable oxygen concentration.  Biochemical pathways of aerotaxis remain largely elusive, however, aerotactic pattern formation is well documented. This allows mathematical modeling to test plausible hypotheses about the biochemical mechanisms.  Our model consists of a system of hyperbolic differential equations, each describing a group of bacteria moving in the same direction, coupled with a diffusion equation describing the dynamics of oxygen. Computer simulations allow comparisons of theoretical and experimental results and help in making a judgment on the validity of various biochemical mechanisms.  Our analytical results give estimates for parameters that are otherwise difficult to obtain experimentally.


Bacterial band: model and experiment


The bottom portion of this  figure shows a band of bacteria formed at the favorable oxygen concentration after 1-3 minutes  oxygen is introduced to the system.  Results of our numerical simulations, shown on top, in which the solid line represents the bacterial density, and the dotted line the oxygen concentration, agree well with the experimental findings. 


Bacterial Pattern Formation II.
Myxobacterial swarming


Although I started this project with  Alex Mogilner, the work was mainly done by him and one of his other students, Angela Gallegos

This work  examines the phenomenon of swarming - spreading of bacterial colony of Myxococcus xanthus on plates coated with a nutrient. The bacteria spread by gliding on the surface. On the time scale of tens of hours, effective diffusion of the bacteria combined with cell division and growth causes a linear increase of the colony s radius at a constant rate. Mathematical analysis and numerical solution of reaction-diffusion equations describing the bacterial and nutrient dynamics demonstrate that the swarming rate is proportional to the square root of the effective diffusion coefficient and nutrient concentration in this regime. In the first few hours, cell division and growth is irrelevant. In this case, bacteria swarm through peninsular protrusions from the edge of the initial colony. We analyze mathematically the diffusion through the narrowing reticulum of cells on the surface and derive formulae for the swarming rates. The model predictions agree with data on the swarming rate dependence on the type of the gliding motility. The model provides a connection between microscopic data on gliding speed and reversal frequencies and macroscopic data on the swarming rates. We discuss the model implications for the cell behavior.



Endothelial Cell Deformation


Endothelial cells go through extensive morphological changes when exposed to shear stress due to blood flow.  These morphological changes are thought to be at least partially the result of mechanical signals, such as deformations, transmitted to cell structures.  Our model describes an endothelial cell as a network of viscoelastic Kelvin bodies with experimentally obtained parameters.  Each part of the cell (each Kelvin body) is described with a linear first order differential equation.  The entire network (representing the cell) is a system of ODEs.   We probe the impact of steady and oscillatory flow on these simple networks.  Matlab simulations of the system and analytical results confirm that steady flow over the network results in much larger deformations than oscillatory flow. Qualitative predictions  about the speed of attaining peak deformation in  different elements representing the flow sensor, nucleus and the cytoskeleton agree with experiments.


Endothelial cell deformation


(A) Time evolution of the deformation of two identical Kelvin bodies connected in parallel in response to steady and oscillatory flow. Because the evolution to the asymptotic response for the two types of flow occurs over different timescales, oscillatory flow evolution is shown in the inset. For both types of flow, a shear force F0 is applied at t = 0. Because they are connected in parallel, the two bodies deform equally. The deformation exhibits an instantaneous jump (at t = 0) due to the elastic springs with subsequent creeping as the dashpot deforms. (B) Time evolution of the force in body 1 in response to steady and oscillatory (inset) flow. (C) Peak deformation of the bodies as a function of the applied shear force in response to steady and oscillatory flow.

Axon Guidance
 
Chemotaxis in animal cells is characterized by a) movement toward attractants and away from repellents; b) ability to detect gradients in a wide range of attractant/repellent concentrations; c) amplification of the external signal  in order to detect even very shallow gradients.   Chemotaxis in the fingerlike protrusions at the tip of axons (growth cones) differs from gradient sensing in other animal cells, because growth cones can change their attractive or repulsive response to the same chemical gradient based on their internal chemical state such as their  calcium or cAMP levels.  We create two models describing different aspects of growth cone guidance.  The first model describes the internal switch that determines the direction of movement.   This model consists of a system of two differential equations (one for calcium, one for a precursor of cAMP) .  The ligand concentration (input to the cell) is a bifurcation parameter.  As this concentration is changed, the system goes through a bifurcation interpreted to be the desired abrupt change in the output of the cellular signaling.  However, in this model the bifurcation is tied to absolute values of the ligand concentration whereas chemotactic cell are known to be sensitive to a wide range of effector concentrations. A second model is created to propose a mechanism that allows growth cone guidance in any environment.  We start with a system of three ODE.  This model exhibits perfect adaptation (the steady state of the output is constant).  If the cell is represented by two compartments, fast-slow analysis shows that the system is able to "adapt" - return to the same steady if the ligand concentration is uniform regardless of the intial conditions,  yet it is able to respond presistently to a spatial gradient.  Analytical work is confirmed by Matlab simulations of the corresponding reaction-diffusion system. 


Dissertation

My dissertation consisted of the three of the above described projects: a mathematical model of aerotactic band formation, a model of endothelial cell deformation and a mathematical model of the signaling underlying axon guidance.  It is rather long.  You can download  it (.ps or .pdf).  (The .ps version has better quality but it's huge.)


Research Presentations


These are some of my research presentations.  They have all been created using Latex  and postprocessing with Ppower4.
 

"Using Math in Cell Biology: A tale of two channel types", (.pdf) Occidental College, April 12, 2007

"Graduate Seminar -- Guest Lecture", HSU, Sep 5, 2006 (Intro.pdf, 10Min.pdf and 30Min.pdf)

"Redwood Empire Mathematics Tournament -- Keynote Address", HSU, March 25, 2006 (.pdf)

"Graduate Colloquium Talk -- Research interests", Graduate Colloquium, Math Dept, HSU, Sep 28, 2005 (.pdf)

"The feedback of a localized calcium domain on calcium-gated channels", Mathematical Biology Seminar, Mathematics Department, University of Utah, Oct 29, 2004. (.pdf)
                                                                                               
"Using Math in Cell Biology: How do Calcium Channels Work?", invited talk, Mathematics Colloquium, Sonoma State University, California, April 21, 2004; Mathematics Colloquium, Humboldt State University, March 25, 2004. (.pdf)
                                                                                               
"Mathematical Model of Bacterial Aerotaxis", Applied Science Seminar, Department of Applied Science, College of William and Mary, Oct. 17, 2002. (.pdf)
                                                                                               
The following posters were also created using Latex. 

"Computational Modeling of Calcium Dynamics Near Heterogeneous Release Sites", (letter size)

"Analysis of the effect of residual Ca2+ on the gating of calcium-regulated calcium channels", poster presentation (letter size or poster size) at the Society for Mathematical Biology Annual Meeting, July 25-28, 2004, Ann Arbor, Michigan and at the Biophys. Soc. Annual Meeting, February 15-18, 2004, Baltimore.



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