Badal Joshi
 

Teaching:

I am teaching Math 135: Probability in Spring 2012.

Here is a link to my teaching from previous semesters.

An important component of Math 132S (Nonlinear Differential Equations) is a project involving a scientific or a mathematical application. Here are the project reports from Spring 2011.

Publications:

• Badal Joshi. Order of magnitude time-reversible Markov chains and characterization of clustering processes. arXiv:1110.5489 [math.PR]. Submitted.

• Badal Joshi and Anne Shiu. Atoms of multistationarity in chemical reaction networks. arXiv:1108.5238 [math.DS]. Submitted.

• Badal Joshi and Anne Shiu. Simplifying the Jacobian Criterion for precluding multistationarity in chemical reaction networks. arXiv:1106.1578 [math.DS]. Accepted for publication at SIAM Journal on Applied Mathematics.
  

• B.Joshi, X.Wang, S.Banerjee, H.Tian, A.Matzavinos and M. Chaplain. On immunotherapies and cancer vaccination protocols: A mathematical modelling approach. Journal of Theoretical Biology Vol. 259, Issue 4, 21 August 2009, pp. 820-827.

• Andrew Gall, Badal Joshi, Janet Best, Virginia R. Florang,  Jonathan A. Doorn and Mark Blumberg. Developmental Emergence of Power-Law Wake Behavior Depends Upon the Functional Integrity of the Locus Coeruleus. SLEEP, Vol. 32, No. 7, 2009.
  

Recent Presentations:

• Atoms of multistationarity in chemical reaction networks, Grad/Fac seminar, (November 18, 2011) Mathematics, Duke University. 
• A Markov chain model for pole formation, SIAM conference on Applications of Dynamical Systems (May, 2011), Snowbird, Utah. (short version)
  
• A Markov chain model for pole formation, Duke Mathematical Biology Colloquium (April, 2011), Mathematics, Duke University. (long version)
  

Clustering and pole formation

Symmetry-breaking, where an initially symmetric state spontaneously changes into an asymmetric state, is a common occurrence in biology. For instance, molecules on the surface of an yeast cell aggregate at one location and form a bud, after which the yeast grows in the direction of the bud. We study a Markov chain model, a combination of two stochastic processes - clustering and diffusion, which gives rise to symmetry-breaking and pole formation. This leads naturally to the notion of order of magnitude reversible Markov chains.

We consider a torus of dimensions (20, 12), initially with alternating vertices either containing one particle or empty. The snapshots are taken at times 1000, 40000, 80000, and 300000.

We present a complete classification by multistationarity of all bimolecular, two-reaction CFSTRs (A reaction network is a CFSTR if all chemical species are in inflow and outflow). A reaction network is 'multistationary' if there exist some positive reaction rates for which the reaction network has multiple steady states under mass-action kinetics. The figure below has a complete list of all multistationary bimolecular, two-reaction CFSTRs. As we move along the arrows, new chemical species are added to the network, but the property of multistationarity is preserved. We identify the roots as 'atoms of multistationarity', because these are minimal CFSTRs (minimal in number of species and in number of reactions) with multistationarity.

Here we display the 35 multistationary bimolecular two-reaction CFSTRs that are minimal with respect to the subnetwork relation.  The poset relation depicted here is that of embedded networks: an arrow points from a network N to a network G if N is an embedded network of G.  In addition, each such edge is labeled by the species that is removed to obtain N from G; for example, C(2) denotes that G contains two molecules of species C, and these two are removed from G to obtain N.  Two networks in the poset are displayed with the same height if they contain the same number of molecules.  The 11 embedded-minimal networks are marked in bold/red; they are the networks that have only outgoing edges in the figure. The figure was created using Mathematica. The figure appears in Atoms of multistationarity in chemical reaction networks.