Bioprocesses are often characterized by nonlinear and uncertain dynamics, posing particular challenges for model predictive control (MPC) algorithms due to their computational demands when applied to nonlinear systems. Recent advances in optimal control theory have demonstrated that concepts from convex optimization, tube MPC, and differences of convex functions (DC) enable efficient, robust online process control. Our approach is based on DC decompositions of nonlinear dynamics and successive linearizations around predicted trajectories. By convexity, the linearization errors have tight bounds and can be treated as bounded disturbances within a robust tube MPC framework. We describe a systematic, data-driven method for computing DC model representations using deep neural networks with a special convex structure, and explain how the resulting MPC optimization can be solved using convex programming. For the problem of maximizing product formation in a cultivation with uncertain model parameters, we design a controller that ensures robust constraint satisfaction and allows online estimation of unknown model parameters. Our results indicate that this method is a promising solution for computationally tractable, robust MPC of bioprocesses.