** Introduction -- Mathematical Modeling in Cell and Molecular Biology, and Relation to Pattern Formation**

A cell is the smallest unit of life that can replicate independently, and cells are often called the ”building blocks of life”. Cell biology is a branch of biology aimed at studying behaviors, functions, structures, and activities of various cells. Cell biology is evolving very rapidly nowadays, and is continuously attracting more and more scientists and researches with various background including mathematics, physics, computer science, statistics and engineering [3, 8]. Although with a great amount of effort addressed on, however, we are still far from a full understanding of the microscopic cellular world.

Patterns, on the other hand, are universal phenomena taking place in physics, chemistry, biology, geography and even economics. For example, spontaneous symmetry breaking (some author also call critical phenomena or non-equilibrium steady states) in second-order phase transition, Belousov-Zhabotinsky reaction, Turing’s pattern, landmark formation and economic oscillations [1, 4, 5, 7]. In the context of biology, there had been lots of studies on pattern formation since Alan Turing’s celebrated work on morphogenesis [9], which had a great influence on revolution of interdisciplinary subjects such as mathematical biology, biophysics and complexity science. However, most of the well-developed theories are mainly focused on macroscopic level biology (e.g. system level biology) so far, and benchmark results are still need in microscopic counterpart (cell and molecular level biology).

Inspired by these facts, my work will thus be focused on developing some theoretical and numerical results on pattern formation phenomena in the context of cell biology. Based on bifurcation theory as well as ordinary and partial differential equations, I’ll further develop theories in [4] and [7], and apply the ideas into some well-known experiments in cell biology.

- William Brock and Anastasios Xepapadeas.
*Pattern formation, spatial externalities and regulation in coupled economicecological systems.*Journal of Environmental Economics and Management, 59(2):149 – 164, 2010. - Radek Erban and Hans G. Othmer.
*From individual to collective behavior in bacterial chemotaxis.*SIAM Journal on Applied Mathematics, 65(2):361–391, 2004. - B. Alberts et al.
*Essential Cell Biology*, 4th edition. 2013. - Michael Field and Martin Golubitsky.
*Symmetry in Chaos*. Oxford University Press, Oxford, UK, 1996. - Andrew Fowler.
*Mathematical Geoscience*. Springer, London, 2011. - James Keener and James Sneyd.
*Mathematical Physiology.*Springer-Verlag New York, Inc., New York, NY, USA, 1998. - Tian Ma and Shouhong Wang.
*Phase Transition Dynamics.*Springer-Verlag New York, 2013. - Iris Meier, Eric J Richards, and David Evans.
*Cell biology of the plant nucleus.*68, 02 2017. - A. Turing.
*The chemical basis of morphogenesis.*Philosophical Transactions of the Royal Society of London B: Biological Sciences, 237(641):37–72, 1952.

** Research Topics **

After a sequence of experiments done by Henry Harris in 1950s, researches
on bacterial chemotaxis has been growing quickly by both mathematicians and
biologists, due to the revolution of modern cell biology, biochemistry and
mathematical biology. Up to now, probably the most well-studied chemotaxis
behavior was on *E. Coli*. Depending on external (or environmental)
conditions, *E. Coli* can be induced to form a variety of spatial
patterns. Such reaction can be simulated by Keller-Segel model, an example
of reaction-diffusion system.

As we all know, however, eukaryotic cells are both structurally and functionally different from bacterial cells, and therefore they employ a different mechanism of chemotaxis. For example, the run-and-tumble strategy adopted by bacterial cells can no longer be observed for eukaryotic cells. Moreover, we expect the chemotactic pattern formed by eukaryotic cells to be different from bacterial cells for this reason.

Besides, with respect to pattern formation in *E. Coli* chemotaxis,
there are still a large of works to be done. The original Keller-Segel model
is far from capturing the full story of *E. Coli* chemotaxis because
it totally neglects the detailed biochemical reaction network involved in
an *E. Coli* cell, and therefore failed to capture the run-and-tumble
nature of an individual cell movement. So we expect more fruitful pattern
types rather than a classical Keller-Segel model can tell us.

A neuron (a nerve cell) is an electrically excitable cell that integrates and transmits information as part of the nervous system. Axons are long slender projections of nerve cells that permit fast and specific electrical communication with other cells over long distances. The ability of nerve cells to extend and maintain these processes is critically dependent on the cytoskeleton.

Microtubules and neurofilaments are both long polymers that align in parallel along the long axis of the axon, forming a continuous overlapping array that extends from the cell body to the axon tip. In healthy axons, microtubules and neurofilaments align along the axon and are interspersed in axonal. However, in many toxic neurodegenerative disorders these two populations of polymers separate from each other, with microtubules and organelles located near the long axis of the axon and neurofilaments displaced to the periphery near the axonal cross-sections. For example, in a sequence of experiments done by Prof. Brown's group in Ohio State University, when they apply a toxic chemical called IDPN (3,3’-iminodiproprionitrile) in physiological saline to adult male rats, a strong microtubule-neurofilament segregation was reported. However, the segregation process is reversible, and wash-out happen in hours.

From the experiments above, we can ask the following questions:

- How do microtubules and neurofilaments segregate each other under abnormal situations?
- Why does the segregation phenomena occur on a time scale of hours and is reversible?
- How is segregation related to impairment of neurofilament transport and axonal swelling?

To answer these questions, we need at first to construct a mathematical model.
We offer two different modeling strategies -- **agent based model ** and
**continuous model **. Agent based model treats every cytoskeleton (microtubule
or neurofilament) as a "agent", and use physical laws to describe their
dynamics:

which is a large system of stochastic differential equations (SDEs). As a comparison, the continuous model takes following form:

which is a coupled nonlocal reaction-diffusion equation. Different parameter
values represent for different situations. For example, after nondimensionalization,
k_{2} = 0.432 corresponds to normal case while k_{2} = 0
corresponds to IDPN-injected case. Numerical simulations indicated a significant
difference between these two situations, which are in accordance with experimental
results:

Normal Case: k |
IDPN-injected case: k |

This project is partially supported by NSF grant #1312966. More details
can be found on my
*candidacy exam presentation*.

- Xue C, Shtylla B, Brown A (2015)
*A Stochastic Multiscale Model That Explains the Segregation of Axonal Microtubules and Neurofilaments in Neurological Diseases.*PLoS Comput Biol 11(8): e1004406. doi:10.1371/journal. pcbi.1004406 - Brown A. In: Pfaff DW, editor.
*Axonal Transport. Neuroscience in the 21st century.*Springer; 2013.

Multiscale modelling challenges occur frequently throughout cell biology and molecular biology, especially in the context of cell migration or cell polarization. To solve multiscale problems, both stochastic and PDE modeling approaches have been well studied and in various of similar context in biology.

Generally speaking, stochastic models are easier to parameterize, can be used to integrate underlying biological phenomena, but hard to analyze mathematically and can be more computational expensive. PDE models, However, are amenable to mathematical analysis and computational methods are well-developed. But on one hand, since various biological processes (typically occurring on different time or length scales) are usually lumped together, it make parameters involved in PDEs difficult to obtain; On the other hand, PDE models are sometimes controversial since they are not able to provide all the details involved in a biological process.

Taking account all those
considerations above, it is important to find underline connections of different
approaches, so that we can make better decisions on **WHEN** and **WHERE**
to use **WHICH ** models.

We thereby propose two different approaches to study multiscale problems:

- Using stochastic hybrid system (similar to the ideas of
*diffusion jump process*,*Levy process*or*hybrid switching diffusions*), to see how stochastic behavior can be approximated by a PDE counterpart. - Using PDE-assisted hybrid methods, to separate our problems into different geometric domains, and apply different strategies on each particular domain. This approach has been well studied in the context of computation.

A sketch of PDE-assisted hybrid algorithm, from Reference [2]

More details of stochastic hybrid system can be found in the last pages
of my
*candidacy exam presentation*, and numerical algorithm of
hybrid methods can be found in my presentation
*PDE-assisted hybrid methods*.

- J. Hu, J. Lygeros, S. Sastry (2000)
*Towards a Theory of Stochastic Hybrid Systems,*HSCC, 2000 - Springer - B. Franz, M.B. Flegg, S.J. Chapman, R. Erban (2013)
*Multiscale Reaction-Diffusion Algorithms: PDE-Assisted Brownian Dynamics,*SIAM J. Appl. Math., 73(3), 1224–1247. (24 pages) - Spill, F., Guerrero, P., Alarcon, T., Maini, P. K., and Byrne, H., 2015,
*Hybrid Approaches for Multiple-Species Stochastic Reaction–Diffusion Models,*J. Comput. Phys., 299, pp. 429–445.