How did the research in axonal transport of proteins metamorphosed into cargo moving in multitracks

This other day, I had been studying Microbiology in the library in the fear of upcoming finals and had gotten really weary of spending a full lovely Sunday with pleasant weather sulking inside the library and as it always happens I started introspecting on my decision to take medicine as my career choice. In concern to that, I googled ‘toughest courses in the world’, just to give myself some empathy, but was disappointed to find that MBBS was not one of them. Instead, I landed up on few very interesting results. I was really mesmerized by one of the courses: Math 55, taught at Harvard as a 2-semester undergraduate course. You should really have a look at it.

In this google semantics search,  I found another interesting course: Mathematical Biology. When I delved more into this seemingly tough, intriguing course of which I had no idea beforehand, I found a compelling article by Avner Freidman on what exactly is mathematical biology and how useful is it.

I am including an engaging excerpt from that article:

Reaction-Diffusion-Hyperbolic Systems in Neurofilament Transport in Axon

Most axonal proteins are synthesized in the nerve cell body and are transported along axons by mechanisms of axonal transport.

A mathematical model was developed by Craciun et al. that determines the profile and velocity of the population of transported proteins, as observed in vivo and in vitro experiments. The model is described by a hyperbolic system of equations

ǫ(∂t + νi∂x)pi = Xn j=1 kijpj for 0 < x < ∞, t > 0, 1 ≤ i ≤ n, where kij ≥ 0 if i ≠ j, Pn i=1 kij = 0 and 0 < ǫ << 1.

Here pi(x, t) is the density of cargo in one of n states (moving forward along a track, moving backward, resting, off track, etc.) and x is the distance from the cell body.

Setting

pm(x, t) = λmQm x − vt √ ǫ , t ,

where λm is determined by the boundary conditions at x = 0 and v is a weighted average of the velocities vi (vi can be positive or negative), it was proved in [9] that

Qm(s, t) → Q(s, t) as ǫ → 0,

where Q(s, t) is the bounded solution of a parabolic system

(∂t − σ 2 ∂ 2 s ) Q(s, t) = 0, −∞ < s < ∞, t > 0,

Q(s, 0) = ( 1 if −∞ < s < 0 q0(s) if 0 < s < ∞;

q0(s) depends on the initial conditions of the pi and σ2 is a function of the kij .

This result, which was inspired by formal calculations in Reed et al., shows that the cargo moves as an approximate wave: its velocity is fixed, but its profile decreases.

The above result was extended to include cargo moving in multitracks. [Avner Friedman & Bei Hu]

 

This article was originally published in American Mathematical Society 2010 Aug 57(10).

The link to the original article: What Is Mathematical Biology and How Useful Is It?

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