Difference between revisions of "Team:Paris Bettencourt/Model"

Abstract

To act on stains, an enzyme must be concentrated at the fabric surface. Our project began with the idea that we can effectively increase this concentration with a fabric binding domain (FBD). But does this idea hold up to detailed scrutiny? What is the optimal affinity for a stain removing enzyme? How much activity improvement can we expect to achieve? To answer these questions, we built three models of the enzyme-fabric-stain interaction: a differential equation model, a stochastic process model and a 3D reaction-diffusion model. The results of these models agreed on three main points. At low affinity, enzymes diffuse away from fabric and into solution. At high affinity, enzymes become trapped on clean sections of fabric and are unable to reach the stain. But for a wide range of conditions, enzymatic activity is optimized at a binding activity around 10^-4 M. For reference, this affinity is much lower than typical antibody-antigen interactions (10^-9 M) and is within reach of our protein engineering methods. Under realistic conditions we predict that an optimal binding domain will improve activity by 100 or 1000 fold.

Motivation and Background

Mass-Action Models

In the simplest model, reactant concentrations are assumed to be uniform throughout the system. We make the mass action assumption: that reaction rates are directly proportional to reactant concentrations. In this way, we can represent the system with just a few simple differential equations. These great advantage of these models is conceptual simplicity and computational speed. It is possible to repeat the simulations many times over large parameter ranges. But it is important to remember that these models do not account for spatial structure. It is as though the fabric has been split into many small pieces and thoroughly mixed.

Stochastic Models

With Gillespie simulations, we are able to keep track of individual molecules. This allows us to account for single molecule dynamics that aren't part of mass action modes. In a mass action model, interactions are averaged over large numbers and continuous, but in a Gillespie model they are discrete.For example, a single enzyme may bind and unbind a stain without ever acting on a stain. A disadvantage of Gillespie simulations is that they can only by computed for very small numbers of molecules. They also do not represent spatial structure.

Diffusion Models

Using the Method of Explicit Fine Differences, we represent the system as a three-dimensional grid of discrete volumes. Within a volume, reactions are modelled as Mass Action, but species may diffuse between volumes at the borders. This allows us to account for the fact that different volumes contain different amounts of enzymes. In particular, we allow the enzyme-fabric binding to concentrate the enzyme near the fabric, where it is most effective. But it is important to remember that we model only passive diffusion, not active mixing. So this type of modelling may tend to overstate spatial effects.

Challenge: Modelling stain removal in a compact washing machine

A typical garment is composed of several square meters of fabric and a typical compact washing machine has a volme of 70 liters.

Results

Model1: Mass-Action Model

Model2: Stochastic Models

Model3: Diffusion Models

Methods

Mass-Action Model

Stochastic Models

Stochastic simulations of our model were performed by applying the Gilespie algorithm. For this, we had to scale down all of the parameters used in the other two models to capture the behavior of single species. The model assumed that cotton cloth consists of the 10 000 fabric binding units arranged in a square 100 by 100. Some of the fabric binding units are next to a stain, so if an enzyme is bound to the particular unit it has the opportunity to clean the stain. We assumed that there are 5% of such stained units, and we normally distributed them in the center of the fabric. We placed 100 enzymes in our system, and observed the the stain degradation by varying kinetic constants; we were specially interested in binding (kon) and unbinding constant (koff). The simulation is captured in the video in which the enzyme dynamics and stain degradation can be observed in real time.

Diffusion Models

Explicit Finite Different scheme is used to model three dimensional stain - enzyme dynamics. The enzyme is assumed to be homogeneously spread through out the spatial domain at the start of the experiment. The scheme was applied on a reaction and diffusion equation thereafter. No flux boundary condition was applied at all boundaries which specifically meant for zero enzyme loss from the system. One of the boundaries is taken as the shirt with stain (1cm^2 area). The parameters and the initial conditions used in the simulations were chosen as realistic as possible. MATLAB was used to computationally model the system and perform thee simulations.

Attributions

This project was done mostly by Mislav Acman and Mani Sai Suryateja Jammalamadaka.

References

• Enzyme Database- BRENDA
• Numerical Analysis and Optimization, An Introduction to Mathematical Modelling and Numerical Simulation- Grégoire Allaire