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Modeling

Theoretical analysis of the process gives us a quantitative insight into the process. We consider a motor having a simplest shape, i.e. a single lamina with a vertical axis in the middle.

Assumptions

An analytical solution can be derived on the basis of the following assumptions:

1. power of a bacteria is kept a constant;
2. friction coefficient of a small object undergoing slow motion is proportional to the cross sectional area;

Let us take a look at the validity of the above assumptions.

A bacterium can be thought of as a truck travelling along an expressway. If it is free of loading and the power is kept constant, the truck will experiences relatively small frictional force and will be moving at a relatively high speed. On the other hand, if the truck is loaded with heavy weights, the frictional force will increase significantly. As a result, the truck will have a much lower maximum speed. However, in both situations, the power of the truck remains the same because of the property of the engine. Same thing happens for bacteria. The power supplied by the motor of the flagella is kept a constant, regardless of the “working condition”, namely whether pushing a motor or not. Thus, the first assumption seems to be reasonable.

The second assumption is taken from the fluid-dynamics, saying that friction is proportional to the cross sectional area for small objects with slow motion. The motor and the bacteria have a size of and respectively, both falling into the region of “small objects”. The low speed of a bacteria, , also fits into the “slow motion” condition. Therefore, the 2nd assumption is believed to be suitable in our case.

Calculation of Rotational Velocity

We start out by calculating the power of a single bacterium first. Under freely swimming condition, a bacterium can move at a maximum speed of (). If the friction coefficient is , then the friction is given by

.

Here, is a function of expression level of CheZ: if CheZ is fully expressed, the bacteria speed should be maximized, correspondingly. On the other hand, if CheZ is completely knocked out, bacteria should lose the swimming ability. Thus, corresponds to this case.

Hence, the power is

.

Next, we estimate the power consumed by the motor, rotating at an angular velocity , due to friction. Consider a small element on the motor from r to r+dr, namely the red part in Fig 2. Let be the friction coefficient of the motor. Hence, from the 2nd assumption, the friction coefficient for the small element is , l here is the width of the motor. Thus, the friction force on this element, proportional to its velocity , is

Power consumed by this element is

The total power consumed on the motor is then the sum, or integration in other words, of all the ,

The power supplied by the bacteria is completely consumed by the motor frictional force. With the conservation of energy, we have

Here, h is the height of the motor along the axis direction and n is the number of bacteria per unit area on the motor.

Resutls and Discussion

According to assumption 2, we can further reduce the above equation. Let , ,

C and are the friction coefficient per unit area of bacteria and motor respectively. a is the cross section area of a bacteria. Substitute them into the above equation gives

here, is just a constant of order .

The model predicts the followings:

1. Angular velocity is independent of height h of the motor;
2. Angular velocity is linearly proportional to the width l of motor and the velocity of bacteria. As a consequence, the expression level of CheZ monotonically affects the rotational velocity. In other words, higher expression level of CheZ results in faster rotation.

This model is only applicable for small objects. So, despite independence of h, the height of the motor still can’t be too large. In other words, it should be confined within . What’s more, even the rotational velocity is inversely proportional to l, the length of the motor can’t be too small because narrow motor would result in too few bacteria attached, which leads to too much noise and fails to fit into this model. To be more precious, a suggested length of motor should be of the order of . Thus, with a motor having a width of , the angular velocity will be about .