Cytomorphic engineering attempts to study the cellular behavior of biological systems using electronics. As such, it can be considered analogous to the study of neurobiological concepts for neuromorphic engineering applications. To date, digital and analog translinear electronics have commonly been used in the design of cytomorphic circuits; Such circuits could greatly benefit from lowering the area of the digital memory via memristive circuits. In this article, we propose a novel approach that utilizes the Boltzmann-exponential stochastic transport of ionic species through insulators to naturally model the nonlinear and stochastic behavior of biochemical reactions. We first show that two-terminal memristive devices can capture the non-linear and stochastic behavior of biochemical reactions. Then, we present the design of several building blocks based on analog memristive circuits that inherently model the biophysical mechanisms of gene expression. The circuits model induction by small molecules, activation and repression by transcription factors, biological promoters, cooperative binding, and transcriptional and translational regulation of gene expression. Finally, we utilize the building blocks to form complex mixed-signal networks that can simulate the delay-induced oscillator and the p53-mdm2 interaction in the cancer signaling pathway. Our approach can provide a fast and simple emulative framework for studying genetic circuits and arbitrary large-scale biological networks in systems and synthetic biology. Some challenges may be that memristive devices with frequent learning and programming do not have the same longevity as traditional transistor-based electron-transport devices, and operate with significantly slower time constants, which can limit emulation speed.