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

Stochastic Models

Diffusion Models

Challenge: Modeling 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.

Figure X: Schema and reaction equations for an ODE model of stain removal. A An enzyme may reversibly bind to clean fabric or stained fabric. For simplicity we assume that these binding constants are equal. Once the enzyme is bound to stained fabric it may be converted to clean fabric. Stain removal is assumed to be irreversible. B The schema gives rise to two binding equilibria and one irreversible reaction. C Three differential equations capture the dynamics of free enzyme, stain-bound enzyme and clean-fabric-bound enzyme. Not shown are similar equations modeling the unbound stained and clean fabric.

Results

Model1: Mass-Action Model

Model2: Stochastic Models

Model3: Diffusion Models

Methods

Mass-Action Model

Stochastic Models

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.

Attributions

This project was done mostly by Jake Wintermute, 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