A computational framework is presented for integrating the electrical, mechanical and biochemical functions of the heart. Finite element techniques are used to solve the large-deformation soft tissue mechanics using orthotropic constitutive laws based in the measured fibre-sheet structure of myocardial (heart muscle) tissue. The reaction-diffusion equations governing electrical current flow in the heart are solved on a grid of deforming material points which access systems of ODEs representing the cellular processes underlying the cardiac action potential. Navier-Stokes equations are solved for coronary blood flow in a system of branching blood vessels embedded in the deforming myocardium and the delivery of oxygen and metabolites is coupled to the energy-dependent cellular processes. The framework presented here for modelling coupled physical conservation laws at the tissue and organ levels is also appropriate for other organ systems in the body and we briefly discuss applications to the lungs and the musculo-skeletal system. The computational framework is also designed to reach down to subcellular processes, including signal transduction cascades and metabolic pathways as well as ion channel electrophysiology, and we discuss the development of ontologies and markup language standards that will help link the tissue and organ level models to the vast array of gene and protein data that are now available in web-accessible databases.