Copyright 2014 by Marquette University and Evgueni Kovriguine

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Equilibrium models

5/13/2011

Contents

1. Introduction

2. Models

3. Notes for developing new models

  Introduction

Here I perform derivations of analytical or numeric expressions for all multi-state systems. A number of models were developed previously in EKM16. I copy those derivations here. I always try to reach analytical solution for equilibrium concentrations. When it is not possible I develop expression for a numerical solution. After derivations are performed I do 'common sense' testing of the models---try to produce predictable behavior of the population graphs in a titration, etc.

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   Models

Location: Mathematical_models/Equilibrium_thermodynamic_models.

Model list

• Original model definitions

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• Intramolecular isomerization (no ligands)

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• One binding site (U family)

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• Mixture of two one-site receptors binding a ligand

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• Two binding sites + additional process (B family)

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• Multi-path U models

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Original model definitions

Original model summaries created in Adobe Illustrator documents and summarized below in PDF.

NOTE: Newest models have their definitions directly in the corresponding model folders.

• Folder: IDAP/Mathematical_models/Equilibrium_thermodynamic_models/Model_definitions
  • Models_1.pdf
  • Models_2.pdf
  • Models_3.pdf
  • Models_4.pdf

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  • B-family
    • B_model_IDAP
    • B-bidentateL-micro _model
    • B_R2_R2L2

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NOTE: For models that are solved numerically---keep L/R>=0.01, otherwise numeric solutions are unstable! Same applies to values of constants if you want to 'turn off' a specific transition: make them small but not too small!

NOTE: Not all models are fully implemented.

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Intramolecular isomerization (no ligands)

I_ab


• 
• Derivation: I_ab_derivation.mn  I_ab_derivation.html

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  • I_ab_T (temperature dependent)
    • This is a derivative model for I-ab so no separate derivation was done.
    • Testing
      • I_ab_T_analysis.mn
      • I_ab_T_analysis.html
      • I_ab_T_testing.m
      • HTML

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I_abcd


• 
• Derivation: I_abcd_derivation.mn         I_abcd_derivation.html  

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• Analysis:    I_abcd_analysis.mn            I_abcd_analysis.html

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One binding site (U family)

U

• 
• Derivation:  U_model_derivation.mn    U_model_derivation.html  (analytical solution)
• Analysis:  U_model_analysis.mn     U_model_analysis.html
• Testing of IDAP implementation: U_testing.m     U_testing.html

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U-L

analytical solution



Derivation: LRIM_U_L.mn    LRIM_U_L.html

Testing of MATLAB implementation:  U_L_testing.m     html/U_L_testing.html

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U-L-RL

Not done yet: the same as U-R-RL---just invert the labels.

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U-L2

numeric solution



Derivation: LRIM_U_L2.mn   LRIM_U_L2.html

Testing of MATLAB implementation:  U_L2_testing.m     html/U_L2_testing.html

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U-R

analytical solution



Derivation:  LRIM_U_R.mn     LRIM_U_R.html

Testing of MATLAB implementation:     

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U-RL

analytical solution - most recent testing of IDAP implementation



Derivation:  LRIM_U_RL.mn      LRIM_U_RL.html    

Testing of MATLAB implementation:    U_RL_testing.m        html/U_RL_testing.html

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U-R-L

analytical solution



U_R_L_derivation.mn    U_R_L_derivation.html

U_R_L_analysis.mn     U_R_L_analysis.html

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U-R-L-RL

 



• U-R-L-RL_derivation.mn    HTML

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• U-R-L-RL_analysis.mn     HTML

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U-R-RL



analytical solution

Derivation: U_R_RL_derivation.mn   HTML

Analysis:    U_R_RL_analysis.mn      HTML

Testing of MATLAB implementation:        U_R_RL_testing.m      HTML

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U-R-RL-RM



• U-R-RL-RM_derivation.mn      U-R-RL-RM_derivation.html
• U-R-RL-RM_analysis.mn       U-R-RL-RM_analysis.html
• U_R_RL_RM_simulations_summary.mn      U_R_RL_RM_simulations_summary.html

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U-R-RL-RM-RLM



• U-R-RL-RM_RLM_derivation.mn       U-R-RL-RM_RLM_derivation.html

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U-R2

numeric solution



Derivation: LRIM_U_R2.mn     LRIM_U_R2.html  

Testing of MATLAB implementation:   see Tutorial 6 tutorial_6_testing_new_model.m     html/tutorial_6_testing_new_model.html

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U-R2L2

numeric solution



Derivation:  LRIM_U_R2L2.mn     LRIM_U_R2L2.html

Testing of MATLAB implementation:     U_R2L2_testing.m     html/U_R2L2_testing.html

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Mixture of two one-site receptors binding a ligand

U_U 



• U__U_derivation.mn       U__U_derivation.html
• U__U_analysis.mn        U__U_analysis.html

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U-R2L2_U



U-R2L2__U_derivation.mn

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Two binding sites + additional process (B family)

B family folder

B (macro), B (micro), and B-bidentateL (micro) 

 

B-macro


B-micro


  most recent numeric solution  

Derivation:  B_family_models_derivation  (B_family_models_derivation.mn)


Analysis: B_family_model_analysis (B_family_model_analysis.mn)

MATLAB implementation

B_macro_testing.m     html/B_macro_testing.html

B_micro_testing.m      html/B_micro_testing.html

B_bidentateL_testing.m       html/B_bidentateL_testing.html

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B_R2_R2L2

Macro


Micro


 


Derivation: B_R2_R2L2_derivation (B_R2_R2L2_derivation.mn)
Analysis: B_R2_R2L2_analysis (B_R2_R2L2_analysis.mn)

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Multi-path U models

• Binding models with multiple binding-competent or incompetent species. Alternative binding species may represent disordered receptor conformations.

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• These models are extension of U-R and U-R-RL models. Since the number of species will be large, whether specific pair of species are interconverting will be defined later - when deriving kinetic matrices. I will only consider equilibrium constants here, for simplicity.

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Generalized schemes

nU-R-RL	U-nR	U-nR-RL

 

U-2R

path: IDAP/Mathematical_models/Equilibrium_thermodynamic_models/U-multi-path-models/nR/2U-R

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• U-2R_derivation.mn     HTML

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• U-2R_analysis.mn       HTML

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• U_2R__testing.m        HTML

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  U-5R

NOTE1: Analytical solution   

NOTE2: IDAP model - - vectorized analytical model

In this model, I am considering five states of unbound receptor incompetent for binding. This is, likely, adequate approximation for "many" states like in case of unfolded proteins. At the same time, I will be defining mass action laws through the isomerization transition B (like in U-2R), which allows to simply turn off unnecessary species by setting their Kb to zero.

path: Mathematical_models/Equilibrium_thermodynamic_models/U-multi-path-models/nR/U-5R



• U_5R_derivation.mn    HTML

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• U-5R_analysis.mn     HTML

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• Introduce U-5R in IDAP - - vectorized analytical model
  • U_5R__testing.m   HTML

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U-2R-RL

This model is a derivative of U-2R

NOTE1: Analytical solution



• U_2R_RL_derivation.mn     HTML

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• U_2R_RL_analysis.mn     HTML

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• Introduce U-2R-RL in IDAP
  • U_2R_RL_testing.mn     HTML

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U-5R-RL

This model is derivative of U-5R

NOTE: Analytical solution   -    most recent analytical solution 



• U_5R_RL_derivation.mn     HTML

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• U_5R_RL_analysis.mn      HTML

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• Introduce U-5R-RL in IDAP
  • U_5R_RL_testing.m     HTML     Vectorized IDAP implementation;

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5U-R-RL

Analytical solution; Vectorized IDAP implementation;

I decided to work with a full 5-branch model, to reduce it later as necessary (as opposed to creating a family of models of increasing complexity). The way to remove unnecessary branches will be to set B1 transition constants to zero (B1 is formation of R' from R*, etc).



• 5U_R_RL_derivation.mn     HTML

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• 5U_R_RL_analysis     HTML

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• IDAP implementation
  NOTE: I am adding 'a' in front of the name because MATLAB cannot use files starting with the number!
  
  
  • a5U_R_RL_testing.m    HTML   Vectorized IDAP implementation;

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   Notes for developing new models

For implementation in IDAP, follow these steps:

• Create a new model file to calculate equlibrium concentrations in code/+equilibrium_thermodynamic_equations/ .
  • If you have analytical solution: create [model-name]_model.m and insert the analytical formula.

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  • If you do NOT have analytical solution: express Rtot=f(Leq, all constants...), where Rtot is the total concentration of a receptor and Leq is equilbrium concentration of free ligand.

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  • Create the model file [model-name]_numeric.m and insert the expression for Rtot.

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  • Create [model-name].m. calling the [model-name]_numeric.m

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    • See example of the numeric solution is U_R2_model.m and corresponding U_R2_model_numeric.m .

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  • Known glitches: at times MuPad stops copying the equations to the clipboard (you don't see an expected copy-paste result). Solution: Relaunch MuPad.

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• Create a test script [model-name]_testing.m in Mathematical_models/Equilibrium_thermodynamic_models/[model-name]
  • see example in   U_RL_testing.m        html/U_RL_testing.html
  • reproduce conditions that were tested when the model was derived to ensure correctness of the model code.

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• Now you can proceed with using this equilibrium model in any other calculations.

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• VECTORIZATION NOTE: Models with analytical solutions may be vectorized to speed up calculations. To vectorize, do this:
  • make sure the Rtotal and LRratio vectors will be supplied as column vectors
  • copy all equations from MuPad into the separate MATLAB script
  • replace all '*' with '.*', '/' with './', and '^' with '.^' in models with analytical solutions (use Replace All function).
  • copy to XXX_model file
  • Convert "Rtotal*LRratio" to "Rtotal.*LRratio"
  • Adjust calls to this vectorized model to read the species concentrations matrix correctly.
  • NOTE: if a non-vectorized model was called as a vectorized one, the error output will look like this
    

    Error using * Inner matrix dimensions must agree. Error in equilibrium_thermodynamic_equations.U_2R_model (line 17) Ltot = Rtotal*LRratio;

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• TITAN implementation note:  use of vectorized models may become essential for the models with large number of transitions, such as 5R or 5U families.

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