Flagellum Propulsion

What is the efficiency of the flagellum?

We are interested in calculating the efficiency of the flagellum as a means of propulsion for the bacteria. Efficiency is defined as the following:

\[\mathcal{E} \equiv \frac{\text{minimum power needed to drag the body}}{\text{power supplied by the motor}} \\ \mathcal{E} = \frac{F_\text{drag}v}{\tau\omega_0} \\ \mathcal{E} = \frac{(Bv)v}{(d\omega)\omega_0}\]

Last lecture we derived 3 equations as well as an approximate form for $v$ which we will use to simplify our expression for \(\mathcal{E}\).

\[\mathcal{E} = \frac{Bv^2}{d\omega_0\left(\frac{B+b}{c}\right)v} \\ \Rightarrow \mathcal{E} = \frac{Bc^2}{d(B+b)^2}\]

Parameters \(B, b, c, d\) are different effective drag coefficients and they all depend on the relative size of the body to the flagellum.

What is the most efficient size regime?

Efficiency depends on relative size. First lets set some shape parameters of the flagellum. Shape parameters depend solely on the characteristics of the flagellum (i.e. helicity, diameter). Let \(\ b_0, c_0, d_0 \\) be these shape parameters. We can relate the shape parameters to the drag coefficients in the following way:

\[b = \alpha b_0 \\ c = \alpha^2 c_0 \\ d = \alpha^3 d_0 \\ \alpha \equiv \text{dimensionless scaling factor}\]

Now we can write \(\mathcal{E}\) in terms of \(\alpha\) and we can take the derivative of \(\mathcal{E}\) with respect to \(\alpha\) to maximize the efficiency.

\[\Rightarrow \mathcal{E} = \frac{B\alpha^4 c_0^2}{\alpha^3 d_0(B+\alpha b_0)^2} \\ 0 = \frac{d\mathcal{E}}{d\alpha} = \frac{Bc_0^2}{d_0} \frac{(B+\alpha b_0)^2 - 2\alpha b_0 (B+\alpha b_0)}{(\text{ . . . })}\]

Simplifying the above equation leads to an expression for \(\alpha^*\), the value for \(\alpha\) at which the efficiency is maximized.

\[\alpha^* = \frac{B}{b_0} \\ \Rightarrow b^* = B\]

We see that the efficiency is maximized when there is equal drag on the body and the flagellum.

\[\mathcal{E}_\text{max} = \frac{c_0^2}{4d_0b_0}\]

This is a very interesting reslt because it does not depend on size! It only depends on the relative shape of the flagellum.

Dynamics of Filament Polymerization

Simple Microtubule Model

In our discussion we will consider a microtubule radiating from a centrosome. Monomers diffusing around the microtubule may bind to the microtubule and become a filament along the length of the tubule. Lets define some quantities of interest in order to describe to growth of the microtubule.

\[c = \text{ concentration of monomers} \\ \alpha = \text{ attach rate} \\ \mu = \text{ detach rate}\]

The concentration, \(c\), has units of number per unit volume. \(\alpha\) and \(\mu\) has units of inverse time. There is an obvious relationship between \(c\) and \(\alpha\), if the concentration goes up so will the attach rate. We will assume that the two quantities are proportional to each other and define the on rate, \(k\), as that factor.

\[\alpha = kc \\ k \equiv \text{ on rate}\]

On the other hand, we will assume that the detach rate is constant and not a function of concentration.

\[\mu \neq f(c)\]

Rate Equation

The rate equation describes the change in the number of filaments \(n\) in the microtubule.

\[\frac{dn}{dt} = \alpha - \mu = kc - \mu\]

As with the case of the detach rate, we will assume that the number of filaments is not a function of concentration.

\[n \neq f(c) \\ \Rightarrow n(t) = (\alpha - \mu)t + n_0\]

Notice that when \(\alpha=\mu\), \(n(t)=\text{constant}\). We can solve for the corresponding concentration of monomers for when this occurs and its known as the critical concentration \(c^*\).

\[c^* = \frac{\mu}{k}\]

\(\frac{dn}{dt}\) Phase Space

If we plot \(\frac{dn}{dt}\) as a function of \(c\) we notice the following:

\[c<c^* \ \Rightarrow \ \frac{dn}{dt}<0 \\ c>c^* \ \Rightarrow \ \frac{dn}{dt}>0\]

This divides the microtubule dynamics into two phases. When \(c<c^*\) the tubule shrinks, and when \(c>c^*\) the tubule grows.

Polymerization at Both Ends

We can also imagine a model where the tubule is not connected to a centrosome on one end. In this case the microtubule can grow and shrink at both ends. We denote one side as the “\(+\)” side and the other as the “\(-\)” side.

\[\frac{dn_+}{dt} = \alpha_+-\mu_+ \\ \frac{dn_-}{dt} = \alpha_--\mu_-\]

In this case there are two critical concentrations, \(c^*_+\) and \(c^*_-\).

Microtubules are known to have roughly the same critical concentrations.

\[c^*_- \approx c^*_+\]

In this case the phase space will be the same as before. When \(c<c^*\) both sides shrink and when \(c>c^*\) both sides grow.

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