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Determining the network of residues that transmit allosteric signals is crucial to understanding the function of biological nanomachines. During the course of a reaction cycle, biological machines in general, and Escherichia coli chaperonin GroEL in particular, undergo large-scale conformational changes in response to ligand binding. Normal mode analyses, based on structure-based coarse-grained models where each residue is represented by an alpha carbon atom, have been widely used to describe the motions encoded in the structures of proteins. Here, we propose a new Calpha-side chain elastic network model of proteins that includes information about the physical identity of each residue and accurately accounts for the side-chain topology and packing within the structure. Using the Calpha-side chain elastic network model and the structural perturbation method, which probes the response of a local perturbation at a given site at all other sites in the structure, we determine the network of key residues (allostery wiring diagram) responsible for the T-->R and R''-->T transitions in GroEL. A number of residues, both within a subunit and at the interface of two adjacent subunits, are found to be at the origin of the positive cooperativity in the ATP-driven T-->R transition. Of particular note are residues G244, R58, D83, E209, and K327. Of these, R38, D83, and K327 are highly conserved. G244 is located in the apical domain at the interface between two subunits; E209 and K327 are located in the apical domain, toward the center of a subunit; R58 and D83 are equatorial domain residues. The allostery wiring diagram shows that the network of residues are interspersed throughout the structure. Residues D83, V174, E191, and D359 play a critical role in the R''-->T transition, which implies that mutations of these residues would compromise the ATPase activity. D83 and E191 are also highly conserved; D359 is moderately conserved. The negative cooperativity between the rings in the R''-->T transition is orchestrated through several interface residues within a single ring, including N10, E434, D435, and E451. Signal from the trans ring that is transmitted across the interface between the equatorial domains is responsible for the R''-->T transition. The cochaperonin GroES plays a passive role in the R''-->T transition. Remarkably, the binding affinity of GroES for GroEL is allosterically linked to GroEL residues 350-365 that span helices K and L. The movements of helices K and L alter the polarity of the cavity throughout the GroEL functional cycle and undergo large-scale motions that are anticorrelated with the other apical domain residues. The allostery wiring diagrams for the T-->R and R''-->T transitions of GroEL provide a microscopic foundation for the cooperativity (anticooperativity) within (between) the ring (rings). Using statistical coupling analysis, we extract evolutionarily linked clusters of residues in GroEL and GroES. We find that several substrate protein binding residues as well as sites related to ATPase activity belong to a single functional network in GroEL. For GroES, the mobile loop residues and GroES/GroES interface residues are linked.