Team:Peking/BindingEvaluation

Introduction

To enhance the effect of killing, we designed and introduced a binding part with which our E. coli could grip the algae with MVN protein. Considering the fact the lysozyme was secreted by our genetically modified E. coli and enriched around the E. coli, the binding to reduce the average spatial distance between the E. coli and algae would potentially promotes the local concentration of lysozyme around the algae, causing the elevation of killing efficiency. A quantitative demonstration is required for this mechanism so we constructed a simulation system using particle collision to mimic the behavior of all essential elements in this system.

We use the collision frequency between lysozymes and algae in simulation system to describe the intensity of killing, similar to the definition of molecular reaction rate. We want to investigate the causality of binding capability and the collision frequency. We are expecting an increase in collision frequency with the rise of binding capability so the binding mechanism could be considered an improvement to the killing.

Method

Our first aim is to demonstrate the efficiency of killing the algae. The killing efficiency, though not thorough enough, could be described by the collision frequency between the algae and lysozymes, the particles that hold killing potential. Since in real situations, the number and distribution of algae varies among situations, we proposed that keeping all numbers and distribution at equilibrium would reflect the behavior of a specific instant.

Our method is basically to use the average of long time equilibrium to illustrate the behavior of the identical parameter conditions. The initial positions of all particles are selected randomly, and the numbers of particles are from the parameters of the specific state. Concrete method is implemented to establish a system where essential elements, including algae, E. coli, and lysozyme, are regarded as rigid spherical particles, and they behave like random walk and collide with each other. The relative collisions and particle positions are calculated and refreshed after a relative short time interval.

Considering the characteristic scale of lysozyme (about 3.6nm), details that how lysozyme collide with water molecules takes little account to our issue, so the movement of lysozyme could be abstract as random walk —a behavior that running a period of time and changing its velocity randomly. To keep this movement conform to the diffusion phenomenon, we also manipulate the length of running time and average velocity so that the distribution varying with time could be in accordance with theoretical result from the Fick Law.

Moreover, E. coli could secrete lysozymes at a constant rate, and to keep the number of lysozyme at the selected equilibrium, we control the lysozyme number by randomly deleting lysozymes in such space.

Since the initial state is selected based on the parameters specific situation, the binding ratio, a predetermined proportion of E. coli bound to the algae, could be an illustration to the binding affinity of such situation. The definition of binding, specifically in this model, is that the E. coli's movements are determined by algae they bound, otherwise simply by their random walk behavior.

Results

Verification of diffusion based simulation

Whether the motion of lysozyme would be influenced by collisions between water molecules and lysozymes should be verified, since unless the influence of water molecules were omitted, the motion of lysozymes could not be simply regard as a diffusion phenomenon. We compared our random walk simulation with theoretical solution under the condition that 5× 10⁴ lysozyme are initialized in the center of a cubic space, and we collected their distribution varying with time. The theoretical distribution was calculated by using the project application model with varied time and space scales. Considering the spatial symmetry, we compared a 1-dimention distribution through the center of the cubic and parallel to axis. The consistency between random walk simulation and diffusion based simulation indicated that the motion of lysozyme could be simplified as a diffusion phenomenon (Fig. 1).

Binding potentially enhances the efficiency of killing

Collision frequency is defined as the count of collision between lysozyme and algae during a time interval, and the collision frequency could be a valid demonstration of killing efficiency. We have tested collision frequency varying with time. We noticed that equilibrium state emerge after a relative short period of time (Fig. 2A). Besides, the result showed a notable distinction between simulations with different binding ratio, which is the proportion of E.coli that adhering to the algae along the simulation (Fig. 2B). There is also a linear relationship between different binding ratios from the parameters of an instant situation, which is defined as the proportion of E. coli binding on algae (Fig. 2C). The slope of the fitting line reflects the certain ratio of binding E. coli caused collisions between lysozymes and algae to the free-moving E. coli caused collisions with the total number of E. coli fixed, and the intercept of the line on Y-axis indicates the basic killing efficiency of all free-moving E. coli. The definition of the slope and Y-axis intercept indicates that the absolute value of the line's X-axis intercept, which equals to the Y-axis intercept divided by the slope, demonstrates the relative ratio of every E. coli's effect on collision frequency at binding state to the E. coli at the free-moving stage. It is shown that the X-axis intercepts of different fitting lines for varied total E. coli numbers are quite similar. Regarding the high value of the slope when the parameter is around the real situation, this similarity further illustrates that the collision frequency could be significantly enhanced by E. coli's binding to the algae.

Collision frequency increases with higher E. coli numbers

Although the simulation above has verified the correlation between binding ratio and collision frequency, however in real situation, it is the number of E. coli binding on algae that finally deterministically influence the collision frequency. We controlled the number of algae and simulated the collision frequency with different numbers of E. coli while the binding ratio remains constant (Fig. 3).

Figure 3. The collision frequency of different E.coli number at a specific binding ratio. The different ratios of algae and E. coli are 0.5, 1 and 2 respectively. The binding ratio of E. coli to algae maintained constantly 0.6 in this figure.

Basal lysozyme collisions without binding has minor effect upon Collision frequency

We also examined the effect of basal concentrations of lysozyme since they were initially given as a constant. Within reasonable variation of lysozyme concentration, we found that raise the concentration of lysozyme could noticeably enhance the collision frequency, but by raise one order of concentration, the enhancement was just about comparable to the enhancement by a small binding ratio of 0.1, when the other parameters were same with conditions in Fig. 1 (Fig. 4).

Figure 4. Comparison between enhancement caused by raising lysozyme concentration and by binding effect. The concentration of lysozyme varies by two orders of magnitude, and there are no E. coli in the simulation for control, and the number in the figure indicates the orders of magnitude of lysozyme with unit of 5× 10₉/L. Black curve indicate the condition, in which the concentration of lysozyme is 5× 10₄ with binding ratio of 0.1 and ratio of algae and E.coli is 1:1.

All results above demonstrate that in the time scale of 1μs magnitude, regional concentration elevation could enhance the killing effect notably proving that introducing binding part is necessary. Furthermore, less lysozyme expression is required to achieve the same effect of killing, thus we could reduce cellular burden. We believe this method of simulating the microscopic particles and cells could have wider implementations in various potential fields, where the different characteristics of particles should be taken into account.