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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.

E. coli is used as the propeller in our project. Glucose is consumed in the mitochondria to generate ATP, which is then used to drive the rotation of flagella. As the number of mitochondria in an E. coli bacterium is kept relatively constant, thus the rate of ATP generation is also constant when the bacteria are provided with excess amount of glucose. This justifies our first assumption that the power of bacteria is a constant.

A bacterium can be thought of as a truck traveling along an expressway. If it is free of loading and the power is kept constant, the truck will experience 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,impeding the movement of the truck. As a result, the truck will have a much lower maximum speed. 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.

The second assumption is based on principles of fluid-dynamics.It is stated that friction is proportional to the cross sectional area of small objects moving at a low speed. The sizes of the motor and the driving bacteria are and respectively, both can be safely regarded as 'small objects'. The low speed of bacteria, which is estimated to be, also satisfies the “slow motion” condition required by the principle mentioned above. Therefore, the second assumption is justified.

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. However, experiments show that polar expression can be either expressed on head or on tail, basically with equal probability. For the ones that have tails expressed, their swimming ability is much reduced when attached to the motor because flagella is physically affected by motor. Therefore, with the concervation 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 .