Problem Calculating equilibrium concentrations from rate constants

Problem Calculating equilibrium concentrations from rate constants 

In many biological problems we know the overall equilibrium constant AND the starting concentrations of the reactants, but we want to know the final concentrations of products and reactants AFTER the equilibrium has been established. A typical reaction is  

Calculate the equilibrium concentrations of receptor, ligand and receptor/ligand complex with the following data  

Assume that all reactions can be described by first order equations. 

In GMA representation the rates for the individual reactants and the product can be written as  

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The computing problem is concerned with calculating the maximum number of unknown parameters of a proposed reaction system from available experimental data. This data can be any combination of values for constant parameters (rate and equilibrium constants) and variable parameters (concentration versus time data). Moreover, data for different variable parameters need not have the same time scale. When the unknown parameters are calculated, it is important that the mathematical validity of the proposed model be determined in terms of the experimental accuracy of the data. Also, if it is impossible to solve for all unknown parameters, then the model must be automatically reduced to a form that contains only solvable parameters. Thus, the input to CRAMS consists of 1) a description of a proposed reaction system model and, 2) experimental data for those parameters that were measured or previously determined. The output of CRAMS is 1) information concerning the mathematical validity of the model and 2) values for the maximum number of computable unknown parameters and, if possible, the associated reliabilities. The system checks for model validity only in those reactions with unknown rate constants. Thus a simulation-only problem does not invoke any model validation procedures. 

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