Team:Peking/BindingEvaluation

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Introduction

To enhance the effect of killing, we designed and introduced a binding part with which our E. coli could grip the cyanobacteria. Considering the fact that lysozyme is 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 cyanobacteria would potentially promote the local concentration of lysozyme around the cyanobacteria, 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 used the collision frequency between lysozymes and cyanobacteria in simulation system to describe the intensity of killing, similar to the definition of molecular reaction rate. We wanted to investigate the causality of binding capability and the collision frequency. We were expecting an increase in collision frequency with the rise of binding capability so the binding mechanism could be considered an improvement to the killing.

Methods

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

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

Considering the characteristic scale of lysozyme (about 3.6nm), details that how lysozymes collide with water molecules takes little account to our issue, so the movement of lysozyme could be abstracted 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 manipulated 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, which keeps the number of lysozymes at the selected equilibrium, we controlled the lysozyme number by randomly deleting lysozymes in such space.

Since the initial state was selected based on the parameters specific situation, the binding ratio, a predetermined proportion of E. coli bound to the cyanobacteria, 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 cyanobacteria they bound to, 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).

Figure 1. Comparison between random walk simulation and diffusion based simulation in different cases: (A) Distribution along the line through the center of the cubic, that is parallel to axis, and (B)(C)(D) indicate the distribution at different instants respectively.

Binding potentially enhances the efficiency of killing

Collision frequency is defined as the count of collision between lysozyme and cyanobacteria 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 cyanobacteria 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 cyanobacteria (Fig. 2C). The slope of the fitting line reflects the certain ratio of binding E. coli caused collisions between lysozymes and cyanobacteria 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 parameters are approximate the real situation, this similarity further illustrates that the collision frequency could be significantly enhanced by E. coli's binding to cyanobacteria.

Figure 2. The simulation result that binding can enhance the killing efficiency. (A) The equilibrium of a specific case could be reached within a short time interval, that after approximate 100 μs from start, the system got into equilibrium state. (B)Higher binding proportion between cyanobacteria and E.coli would cause the increase of collision frequency. In this figure, we selected binding ratio of 1.0, 0.8, 0.6, 0.4, 0.2 and 0. (C) The approximate linear relations of average collision frequency of different binding ratio in (B), where the X-axis intercept indicates the relative ratio of contribution to the collision between lysozymes and cyanobacteria by binding E. coli to the free-moving E. coli.

Collision frequency increases with higher E. coli population

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 cyanobacteria that finally deterministically influence the collision frequency. We controlled the number of cyanobacteria 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 cyanobacteria and E. coli are 0.5, 1 and 2 respectively. The binding ratio of E. coli to cyanobacteria 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 cyanobacteria 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 assay simulating the microscopic particles and cells could have wider implementations in various fields, where the different characteristics of particles should be taken into account.