Bayesian Optimization for Drug Candidate Design

This project implements a Bayesian Optimization (BO) framework to optimize the structural features of a drug candidate. The goal is to maximize bioavailability $logP$ while ensuring the synthesizability constraint $SA < \kappa$ is satisfied.

Problem Formulation

Let $x \in [0, 10]$ represent the structural features. The optimization problem is formalized as:

$$x^* = \arg\max_{x \in X, \, v(x) < \kappa} f(x)$$

where:

• $f(x)$: Bioavailability ($logP$).
• $v(x)$: Synthesizability score ($SA$).
• $\kappa = 4$: Maximum allowed synthesizability.

The observed values are noisy:

$$y_f = f(x) + \epsilon_f, \quad \epsilon_f \sim \mathcal{N}(0, \sigma_f^2)$$ $$y_v = v(x) + \epsilon_v, \quad \epsilon_v \sim \mathcal{N}(0, \sigma_v^2)$$

where $\sigma_f = 0.15$ and $\sigma_v = 0.0001$.

An example problem looks as follows:

Approach

Gaussian Process Modeling

Two Gaussian Processes (GPs) are used:

1. For $f(x)$: Bioavailability
   • Kernel options: Matérn ($\nu = 2.5$) or RBF.
   • Tunable parameters: Variance and length scale.
2. For $v(x)$: Synthesizability
   • Kernel: Additive (Linear + Matérn or RBF).
   • Prior mean: $4$.

Acquisition Functions

The algorithm implements three acquisition functions for constrained optimization:

1. Upper Confidence Bound (UCB):

   $\text{UCB}(x) = \mu_f(x) + \beta \cdot \sigma_f(x) - \max(0, \mu_v(x) - \kappa)$

2. Constrained Expected Improvement (CEI):

   $\text{CEI}(x) = EI(x) \cdot P(v(x) &lt; \kappa)$

   where $EI(x)$ is the Expected Improvement.

3. Probability of Feasibility (PoF):

   $\text{PoF}(x) = \Phi\left(\frac{\kappa - \mu_v(x)}{\sigma_v(x)}\right)$

   where $\Phi$ is the CDF of the standard normal distribution.

Constraints and Safe Initialization

• The optimization starts from an initial safe point $x_\triangle$ where $v(x_\triangle) &lt; \kappa$.
• Unsafe evaluations ($v(x) \geq \kappa$) are penalized.

Results and Evaluation

Metrics

The algorithm's performance is evaluated using normalized regret:

$$r_j = \max\left(\frac{f(x_\bullet) - f(\tilde{x}_j)}{f(x_\bullet)}, 0\right)$$

where $x_\bullet$ is the local optimum and $\tilde{x}_j$ is the suggested solution.

The final score combines regret, penalties for unsafe evaluations, and trivial solutions:

$$S_j = 1 - 0.4 \cdot r_j - 0.6 \cdot h(3N_j - 3.25) - 0.15 \cdot I[|\tilde{x}_j - x_\triangle| \leq 0.1]$$

where $N_j$ is the number of unsafe evaluations and $h(\cdot)$ is the sigmoid function.

• Final Score: 0.863 (ranked 90/273)

Visualization

The algorithm generates (here):

1. Posterior plots of $f(x)$ and $v(x)$, including uncertainty bounds.
2. Acquisition function values for each iteration.
3. Safe and unsafe evaluation regions.