There is growing awareness of the need for mathematics and computing to quantitatively understand the complex dynamics and feedbacks in the life sciences. Although individual institutions and research groups are conducting pioneering multidisciplinary research, communication and education across fields remains a bottleneck. The opportunity is ripe for using education research principles to develop new mechanisms of cross-disciplinary training at the intersection of mathematics, computation and biology. In this paper we present a case study which describes the efforts of one computational biology lab to rapidly prototype, test, and refine a mentorship infrastructure for undergraduate research experiences in alignment with the computational apprenticeship theoretical framework. We describe the challenges, benefits, and lessons learned, as well as the utility of the computational apprenticeship framework in supporting computational/math students learning and contributing to biology, and biologists in learning computational methods. We also explore implications for undergraduate classroom instruction, and cross-disciplinary scientific communication.