Mechanisms for facilitated target location and the optimal number of molecules in the diffusion search process

We investigate the number N of molecules needed to perform independent diffusion in order to achieve bonding of a single molecule to a specific site in time t(0). For a certain range of values of t(0), an increase from N to kN molecules (k>1) results in the decrease of search time from t(0) to t(0)/k. In this regime, increasing the number of molecules is an effective way of speeding up the search process. However when N> or =N0 (optimal number of N) the reduction of time from t(0) to t(0)/k can be achieved only by an exponentially large increase in the number of molecules [from N to N exp(ck) for some c>0].