We examine the role of self-maintenance (collective autocatalysis) in the evolution of computational biochemical networks. In primitive proto-cells (lacking separate genetic machinery) self-maintenance is a necessary condition for the direct reproduction and inheritance of what we here term Cellular Information Processing Networks (CIPNs). Indeed, partially reproduced or defective CIPNs may generally lead to malfunctioning or premature death of affected cells. We explore the interaction of this self-maintenance property with the evolution and adaptation of CIPNs capable of distinct information processing abilities. We present an evolutionary simulation platform capable of evolving artificial CIPNs from a bottom-up perspective. This system is an agent-based multi-level selectional Artificial Chemistry (AC) which employs a term rewriting system called the Molecular Classifier System (MCS). The latter is derived from the Holland broadcast language formalism. Using this system, we successfully evolve an artificial CIPN to improve performance on a simple pre-specified information processing task whilst subject to the constraint of continuous self-maintenance. We also describe the evolution of self-maintaining, crosstalking and multitasking, CIPNs exhibiting a higher level of topological and functional complexity. This proof of concept aims at contributing to the understanding of the open-ended evolutionary growth of complexity in artificial systems.