Problem X.2: Mechanisms for morphogen gradient scaling¶
Stated mathematically, scaling of a morphogen gradient means that the observed length scale of the gradient $\lambda$ is a linear function of the length of the tissue $L$. Another way of stating this is that the dynamics are only dependent on $\tilde{x} \equiv x/L$, at least over a substantial portion of the domain $0 \le x \le L$.
If you think in the context of evolution, mechanisms that produce gradients that scale might be more apt to survive selective pressure than those that do not. For example, if the size of the imaginal wing disc grows, if the gradient does not grow proportionately, then the wing will be disproportionate, possibly with more veins positioned toward the anterior of the wing. In this problem, we will explore mechanisms for gradient formation and investigate under what regimes they may exhibit scaling. We will consider one-dimensional models in all cases. We will denote morphogen concentration as $c(x, t)$.
a) In 1970, Francis Crick proposed a simple mechanism for formation of a morphogen gradient. He postulated that a source of morphogen might exist at position $x=0$ and a sink at position $x=L$. To clarify what Crick means by "sink," we'll use his own words.
It is particularly easy to make a sink, if the sink holds the concentrations of the morphogen near zero, since then all that is required is an enzyme in the sink cells to destroy the morphogen very rapidly, even at very low concentrations.
i) Derive an expression for the steady state profile of morphogen for this model.
ii) Does the steady state distribution scale? If not, what can be modulated to make it scale?
b) Suppose that all of the cells in the tissue being patterned produce a diffusible mobilizing molecule (with concentration $m(x,t)$ and diffusion coefficient $D_m$) at constant rate $q$ (in units of concentration of unit time). The mobilizer affects the diffusion coefficient of the morphogen as
\begin{align} D = D_0\,f(m), \end{align}where $f(m)$ is some function of the mobilizer concentration, $m$. We can write $f(m)$ as a Taylor series to first order, $f(m) \approx 1 + a m$. We assume that the mobilizer has no effect on the reaction rates involving the morphogen. So, the complete reaction-diffusion equation for the morphogen concentration $c(x,t)$ is
\begin{align} \frac{\partial c}{\partial t} = \frac{\partial}{\partial x}\left(D_0(1+a m)\,\frac{\partial c}{\partial x}\right) + r(c). \end{align}Suppose further that there are sinks for this mobilizer on each end of the tissue. That is, $m(x=0) = m(x=L) = 0$, where $m(x)$ is the concentration of mobilizer, as Crick proposed.
i) Solve for the mobilizer concentration, $m(x)$.
ii) Provided $r(c)$ does not have any strange $L$-dependence, does the morphogen profile scale? More specifically, does it scale exactly, approximately, or not at all? Discuss in which limits scaling might be most effective.
c) The concept of a globally secreted molecule that affects diffusion can be analogously applied to one that affects the reaction rate. Imagine that instead of the mobilizer from part (b), an inhibitor is secreted. That is, it inhibits the rate that reactions involving the morphogen can occur, for example by transiently binding it to protect it. Approximating this slowdown as a first order inhibiting Hill function, the reaction-diffusion equation for the morphogen is
\begin{align} \frac{\partial c}{\partial t} = D\,\frac{\partial^2 c}{\partial x^2} + \frac{r(c)}{1 + b h}, \end{align}where $h$ represents the concentration of the reaction inhibitor. If the inhibitor has analogous dynamics as the mobilizer from part (b), does the steady state profile scale? Discuss appropriate limits.