Team:Peking/firsttry/modeling/binding evaluation

Introduction:

To enhance the effect of killing, we designed and introduced a binding part (hyperlink) with which our E. coli could grip the algae with MVN protein so that regional level of lysozyme around algae would go up, which secreted by our E. coli (hyperlink). To demonstrate this mechanism quantitatively, we construct a system consisting of all essential elements.

Like reaction rate of molecules, we use collision frequency between lysozymes and algae in the simulation system to describe the intensity of Killing. Thus if there is a major distinction between systems whose E. coli carries binding part and systems whose not, we could say that binding part remarkably enhance the killing effect.

Method:

First, if we want to calculate the collision frequency between lysozymes and algae at a particular instant, at which all elements in the system have definite number and distribution, consequently, definite collision frequency. But in the real system, number and distribution of all elements vary from time to time, the collision frequency is inconstant. Then if we want to measure the collision frequency at a particular moment, we have to keep all number and distribution equilibrium.

To do that, we give a random initial distribution of all particle, and keep the number of particles respectively. After a period of time, the state of the system becomes equilibrium, and we use that state to simulate reality. Then we count the time of collision between lysozymes and algae in a period of time so that collision frequency is available.

Concrete method is to establish a system in which essential elements--Algae, E. coli, lysozyme are abstracted into rigid spherical particles, which behave random walk and collision with each other. E. coli have ability to secrete lysozymes, and to keep the number of lysozyme without destroy the distribution, we delete lysozyme randomly in the whole space every step.

Initialization

To control binding state, we set a binding ratio at start and will not change during a single program operation. Binding ratio means the percentage that number of E. coli binding on algae account for. The movement of these E .coli is determined by algae they bound. All algae, "free" E. coli, basal lysozyme are given a random position in the simulation space. Their velocities are determined by their diffusion coefficient.

Random walk

Considering the characteristic scale of lysozyme (about 3.6nm), details that how lysozyme collide with water molecules makes little difference to the issue that we care about. So the movement of lysozyme could be abstract as random walk--running a period of time and changing its velocity randomly. To keep this movement conform to the diffusion phenomenon, we manipulate the length of running time and average velocity so that the distribution varying with time could be in accordance with theoretical result which is calculated by Fick Law.

Result:

Space distribution of lysozyme in the simulation in which their motion is abstracted by random walk shows coherence with theoretical solution of diffusion equations. In the program, 50000 particles are placed in a 1μm×1μm×1μm cubic space located at the center of the whole simulation space. Considering the symmetry, we just picked up the concentrations on the line through the central point and parallel to X-axis and compared them to theoretical results. Compares at different instants have been made and they all show effective results. A little fluctuation in center could be explained as a result of discrete method.

And the measurement of collision frequency also shows distinction between conditions with binding and without. Collision frequency varying with time is showed in figure2, in which we could see that there is an obvious positive correlation between binding ratio and collision frequency. To eliminate the effect of basal concentration, we also do control test that there are no E. coli and result shows that the effect is small. Basal concentration selection was referred to out killing experiment, and to avoid exception, 100 times higher concentration is also tested, in which the collision caused by basal concentration still could not reach the level of binding condition.

Well, results above demonstrate that introducing binding part will enhance killing effect dramatically. In another word, to achieve the same effect, less energy is required to express and secrete lysozyme when binding part is introduced. We can also conclude from this model that higher regional concentration will do better than same average but well-distributed, which would be universal when solving similar problems.