Team:Johns Hopkins/Modeling/Methods

Methodology

Engineering a metabolic pathway requires the introduction of computational methods with the aim to characterize, understand and design an efficient synthetic circuit. The design of the VitaYeast pathways follows a well established engineering workflow (e.g. electrical engineering), which is made of three steps:

• Modeling
• Sensitivity Analysis
• Optimization

Modeling

The modeling step is demanded to describe the behavior of a given pathway, in-silico; in computational terms, the model can be viewed as a black-box, which takes in input some initial conditions and output the time-course behavior of the pathway. This abstraction assumptions is central in our design flow, and it permits to establish a generic and scalable methodology.

Sensitivity Analysis

Sensitivity Analysis (SA) tries to correlate the uncertainty of the output with the variation of input conditions; in particular, the goal is to elucidate parameters that mostly influence the output of the model. This step has a twofold implication; from a biological point of view, it is important to understand the most important parts of our pathway, whereas, from a computational point of view, the identification of key parameters permits the reduction of the free variables in the system and, consequently, reduces the number of possible scenario to check (the search landscape).

In our analysis, we represent the this as \[d[vitamin]/d[parameter_i]\] We normalize these sensitivities in two ways. First, we non-dimensionalize them by dividing by the vitamin concentration and multiplying by the starting value of the parameter. Then we obtain relative sensitivities by scaling them so the maximum sensitivity is 1.

We noticed that sensitivity to the Michaelis-Menten binding constant k_m is nearly zero in all pathways. We can interpret this as meaning the substrate concentration is high enough to saturate the enzyme at each step of the pathway. V_max, which is the product of k_cat, and the enzyme concentration, controls the reaction rate. Thus in our analyses we show the sensitivities with respect to k_cat.

Vitamin A | Vitamin C

Optimization

Reducing the biological behavior of a pathway to a function, characterized by specific input/output parameters, represents the classical scenario where optimization methods can be applied; for example, finding the enzymes concentrations that maximizes the production of a certain vitamin. Since the pathway is modeled as a black-box function, we adopt optimization methods belonging to the class of 'derivative-free' algorithms; these methods relies only on the objective function value to find a near-optimal solution to an optimization problem.

Parameter fitting

While a model takes known qualitative information and organizes it systematically, parameter-fitting allows us to compare models with measured data, giving our models predictive power. Parameter fitting can be construed as an optimization problem. Here, we attempt to minimize an error function between our model and our observations. A common error function is the squared error:

\[||simulation-observation||{_{2}}^{2}=\sum_{i,j}^{N}(simulation_{i})^{2}-(observation_{j})^{2}\]

While mathematically elegant, it does not explicitly bound how badly a particular point might be fit. To ensure that even our worst points are reasonably well-fit by the model, we use the minimax error function:

\[max_{i,j}[(simulation_{i})^{2}-(observation_{j})^{2}]\]

In addition using the SimBiology Toolbox in Matlab, we have access to a convenient graphical interface for parameter fitting. Matlab uses the Levenberg–Marquardt Algorithm (LMA), which minimizes the least-squares error function mentioned above. While fairly robust, LMA has two drawbacks: it finds only a local minimum of the error function, and it assumes no measurement error in the measurements parameters. Nonetheless, it is a popular method that works well on most models.

We are currently awaiting the result of experiments to quantify the production of vitamins and other products using spectroscopy and HPLC. Once the data is available, our modeling group will be able to format the data and fit our model using LMA and minimax optimization.

Multi-Objective Optimization

In biology, like engineering, it is not rare to require the attainment of different and, possibly, conflicting objectives; for example, reaching a place as fast as possible with minimum fuel consumption. Multi-objective optimization aims to identify the largest set of trade-off solutions, using the well known financial principle of Pareto optimality; in these terms, a set of Pareto optimal solutions is made of all the feasible solutions where an improvement in any objective causes a worsening of another objective. An easy way to represent this concept are Pareto frontiers plots, which represent a graphical representation of the Pareto optimal solutions in a D dimensional space, where D is the dimension of the problem.