Conceptual models of the atom preceded the mathematical model of the hydrogen atom in physics in the second decade of the 20th century. The computer modeling of the living cell in the 21st century may follow a similar course of development. A conceptual model of the cell called the Bhopalator was formulated in the mid-1980s, along with its twin theories known as the conformon theory of molecular machines and the cell language theory of biopolymer interactions [Ann. N.Y. Acad. Sci. 227 (1974) 211; BioSystems 44 (1997) 17; Ann. N.Y. Acad. Sci. 870 (1999a) 411; BioSystems 54 (2000) 107; Semiotica 138 (1-4) (2002a) 15; Fundamenta Informaticae 49 (2002b) 147]. The conformon theory accounts for the reversible actions of individual biopolymers coupled to irreversible chemical reactions, while the cell language theory provides a theoretical framework for understanding the complex networks of dynamic interactions among biopolymers in the cell. These two theories are reviewed and further elaborated for the benefit of both computational biologists and computer scientists who are interested in modeling the living cell and its functions. One of the critical components of the mechanisms of cell communication and cell computing has been postulated to be space- and time-organized teleonomic (i.e. goal-directed) shape changes of biopolymers that are driven by exergonic (free energy-releasing) chemical reactions. The generalized Franck-Condon principle is suggested to be essential in resolving the apparent paradox arising when one attempts to couple endergonic (free energy-requiring) biopolymer shape changes to the exergonic chemical reactions that are catalyzed by biopolymer shape changes themselves. Conformons, defined as sequence-specific mechanical strains of biopolymers first invoked three decades ago to account for energy coupling in mitochondria, have been identified as shape changers, the agents that cause shape changes in biopolymers. Given a set of space- and time-organized teleonomic shape changes of biopolymers driven by conformons, all of the functions of the cell can be accounted for in molecular terms-at least in principle. To convert a conceptual model of the cell into a computer model, it is necessary to represent the conceptual model in an algebraic language. To this end, we have begun to apply the process algebra of Milner [Communicating and Mobile Systems: The pi-calculus, Cambridge University Press, Cambridge, 1999] to develop what is here called the "shape algebra," capable of describing complex and mobile patterns of interactions among biomolecules leading to cell functions.