- To better understand the enzyme-stain-fabric system.
- To choose an optimal binding affinity for our fabric binding domain.
- To predict how much enzyme activity improvement we can expect.
- Three model classes with diverse assumptions.
- Enzyme activity can be optimized at intermediate binding constants near 104 M-1.
- We predict a 100 fold improvement from an optimzed enzyme
- ODE solvers
- Gillespie Algorithm
- Explicit Finite Difference method (FDM)
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 ModelsIn 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 ModelsWith 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 ModelsUsing 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.
Figure 1 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.
Tabel 1: Key parameters for modeling the enzyme-fabric-stain system. Our model attempts to simulate the case of a white t-shirt stained by a single drop of red wine. The reactions take place within a standard 70 L washing machine. The quantity of enzyme in the system was taken from commercial laundry detergents, which typically contain about 1% enzymes by mass. We take the enzyme CatA as an example, taking its catalysis rate from the BRENDA enzyme database.
|Volme of a washing machine||70 L|
|Volume of a wine stain||50 μL|
|Malvidin concentration in wine||200mg/L|
|Area of cotton t-shot||150000cm^2|
|Weight of cotton||20mg/cm^2|
|Total mass of detergent||5g|
|Enzyme fraction in detergent||1%|
|Activity of CatA enzyme||200s^-1|
Model1: Mass-Action Model
Figure 2 Computational analysis using Mass-Action model. A The three axis 2D plot consisting of KD, Stain fraction and stain remaining after 90mins of enzyme degradation. Enzymatic activity is optimized at a binding activity around 10-4 M. B Typical parameters that need to be considered for the computational analysis, the optimum KD is much lower than typical antibody-antigen interactions (10-9 M).
Figure 3 Stain and bound enzyme percentages estimated using Mass-Action model. A The optimum KD for stain removal is found to be 10-4 when stain percentage is plotted against the time of degradation. B The percentage of different forms of enzyme present in the solution such as free form, enzyme-clean fabric complex, enzyme-dirty fabric complex as a function of KD is shown.
Model2: Stochastic Models
Figure 4 A Stochastic model of enzyme binding was constructed using near-optimal parameters, as determined by the mass-action model. The KD is set to 10-6 and the stain is assumed to cover a 5% of the fabric. EBc represent senzymes bound to clean fabric; EBd is enzymes bound to dirty fabric and E is free enzymes. Under these conditions, almost all of the enzymes are fabric-bound. Because Gillespie simulations play out with few molecules and small scale, this stain is cleaned much faster than in the mass action model above.
Model3: Diffusion Models
Figure 5 Simulations in the diffusion model agree with the mass-action model under most parameters. When diffusion is accounted for, the enzyme activity reaches a maximum near a KD of 10-3. This is a weaker affinity than predicted by mass action alone, perhaps indicating that diffusion is increasing the effective enzyme concentration.
ODE simulations were performed in MATLAB using the ode23t solver and the indicated parameters.
Gillespie's algorithm was performed numerically in MATLAB. Parameters from the mass action model were scaled down to accomodate a countable number of single molecules. The cloth is modeled as grid of binding sites 100 units square, of which 5% are marked as stained. 100 free enzymes are introduced to the system at time zero and the reaction continutes until all the stains are removed.
The Explicit Finite Difference Method was perfomed numerically in MATLAB. The enzyme was initially assumed to be homogenously spread in the spatial domain. A no flux boundary condition was applied, preventing enzymes from leaving the system. One of the boundaries is simulated as a shirt of 1 m2 ares with a 1 cm2. Other parameters were chosen as in the mass action model.
This project was done mostly by Mislav Acman and Mani Sai Suryateja Jammalamadaka.
- Enzyme Database- BRENDA
- Numerical Analysis and Optimization, An Introduction to Mathematical Modelling and Numerical Simulation- Grégoire Allaire