es_sample_elasticities

 ES_SAMPLE_ELASTICITIES - Elasticity phase

 result = es_sample_elasticities(N, W, ind_ext, es_constraints, es_options, c0, c, u, J, Keq, mu, K, Kma, zeta, A)

 Elasticity sampling: run elasticity sampling phase and MCA phase 


 Inputs (= Outputs of es_sample_steady_state; with nm: # metabolites; nr: # reactions)

   N         - Stoichiometric matrix (nm x nr)
   W         - Allosteric regulation matrix (nr x nm)
   ind_ext   - indices of external metabolites
   c0        - Vector of chemical equilibrium concentrations (optional)
   c         - Vector of concentrations
   u         - Vector of enzyme levels
   J         - Flux vector
   Keq       - Vector of equilibrium constants
   mu        - Vector of chemical potentials
   K         - Kernel matrix 
   Kma       - Vector of mass-action ratios
   zeta      - zeta vector computed from reaction affinities
   A         - Vector of reaction affinities (thermodynamic driving forces)

 For the inputs 'es_constraints' and 'es_options', see 'es_default_options'

 Fields in 'es_constraints' and 'es_options' relevant to this function:

   es_constraints.beta_M_fix     - predefined beta values (only used if non-zero)
   es_constraints.beta_A_fix     - predefined beta values (only used if non-zero)
   es_constraints.beta_I_fix     - predefined beta values (only used if non-zero)
   es_options.kinetic_law        - Rate law {'cs','ms', ...}
   es_options.h                  - vector of Hill exponents (one for each reaction)
   es_options.no_equilibrium     - flag: assume no equilibrium for inactive reactions
   es_options.KV_prior_mean      - .. and KV value to be used in this case
   es_options.flag_test          - Run tests? (Boolean)
   es_options.flag_second_order  - Compute second-order elasticities? (Boolean)
   es_options.zv                 - Flux weights for 2nd order resp. coeff output function
   es_options.zc                 - Conc. weights for 2nd order resp. coeff output function

 All output data are stored in a structure 'result' with fields:
   result.c            - Metabolite levels      
   result.c0           - Equilibrium metabolite levels      
   result.u            - Enzyme levels
   result.A            - Reaction affinities
   result.zeta         - zeta vector (computed from reaction affinities)
   result.mu           - Chemical potentials
   result.J            - Fluxes
   result.v            - Reaction rates (computed from rate laws, v=J)
   result.v_plus       - Microscopic forward rates
   result.v_minus      - Microscopic reverse rates
   result.Keq          - Equilibrium constants
   result.Kma          - Mass-action ratios
   result.KA           - Activation constants
   result.KI           - Inhibition constants
   result.KM           - Reactant constants
   result.KV           - Velocity constants (geometric mean of catalytic constants)
   result.Kplus        - Forward catalytic constants
   result.Kminus       - Reverse catalytic constants
   result.saturation   - Saturation values
   result.elasticities - Elasticities (scaled and unscaled)
   result.control      - Control and response coefficients (scaled and unscaled)
   result.K            - Kernel matrix 
   result.kinetics     - kinetic rate laws (to be inserted into 'network')

 See also: es_sample_model