On minimizing the maximal characteristic frequency of a linear chain

We consider a linear chain of masses, each coupled to its two nearest neighbors by elastic springs. The maximal characteristic frequency of this dynamical system is a strictly convex function of certain parameters that depend on the masses and spring elasticities. Minimizing the maximal characteristic frequency under an affine constraint on these parameters is thus a convex optimization problem. For a homogeneous affine constraint, we prove that the mass and elasticity values that minimize the maximal characteristic frequency have a special structure: They are symmetric with respect to the middle of the chain and the optimal masses [spring elasticities] increase [decrease] toward the center of the chain. Intuitively speaking, this means that in order to minimize the maximal characteristic frequency we need to 'fix' the center of the chain, by increasing [decreasing] the masses [spring elasticities] there. We further show that minimizing the maximal characteristic frequency of the linear chain is equivalent to maximizing the steady-state protein production rate in an important model from systems biology called the ribosome flow model.