steadyStateEquations -- creates steady-state equations of a reaction network

Description

Obtain the steady-state equations for a given network. Before any equations can be created, you must invoke the createRing function, which creates the reaction network ring. If you do not create the ring, you will receive an error message.

i1 : N = reactionNetwork "A <--> 2B, A + C <--> D, B + E --> A + C, D --> B+E"

o1 = A-->2B
     2B-->A
     A+C-->D
     D-->A+C
     D-->B+E
     B+E-->A+C

o1 : ReactionNetwork

i2 : R = createRing(N, QQ)

o2 = R

o2 : PolynomialRing

i3 : steadyStateEquations N

o3 = | xx_B^2kk_{1, 0}-xx_Axx_Ckk_{2, 3}+xx_Bxx_Ekk_{4, 2}-xx_Akk_{0, 1}+xx_
     | -2xx_B^2kk_{1, 0}-xx_Bxx_Ekk_{4, 2}+2xx_Akk_{0, 1}+xx_Dkk_{3, 4}     
     | -xx_Axx_Ckk_{2, 3}+xx_Bxx_Ekk_{4, 2}+xx_Dkk_{3, 2}                   
     | xx_Axx_Ckk_{2, 3}-xx_Dkk_{3, 2}-xx_Dkk_{3, 4}                        
     | -xx_Bxx_Ekk_{4, 2}+xx_Dkk_{3, 4}                                     
     ------------------------------------------------------------------------
     Dkk_{3, 2} |
                |
                |
                |
                |

             5      1
o3 : Matrix R  <-- R

Obtain the steady-state equations for a motif and display equations.

i4 : N = modificationOfTwoSubstratesI()

o4 = S_0+E-->X_1
     X_1-->S_0+E
     X_1-->E+S_1
     S_1+F-->Y_1
     Y_1-->S_1+F
     Y_1-->S_0+F
     E+P_0-->X_2
     X_2-->E+P_0
     X_2-->E+P_1
     F+P_1-->Y_2
     Y_2-->F+P_1
     Y_2-->F+P_0

o4 : ReactionNetwork

i5 : R = createRing(N, QQ)

o5 = R

o5 : PolynomialRing

i6 : steadyStateEquations N

o6 = | -xx_S_0xx_Ekk_{0, 1}+xx_X_1kk_{1, 0}+xx_Y_1kk_{4, 5}                  
     | -xx_S_0xx_Ekk_{0, 1}-xx_Exx_P_0kk_{6, 7}+xx_X_1kk_{1, 0}+xx_X_1kk_{1, 
     | xx_S_0xx_Ekk_{0, 1}-xx_X_1kk_{1, 0}-xx_X_1kk_{1, 2}                   
     | -xx_S_1xx_Fkk_{3, 4}+xx_X_1kk_{1, 2}+xx_Y_1kk_{4, 3}                  
     | -xx_S_1xx_Fkk_{3, 4}-xx_Fxx_P_1kk_{9, 10}+xx_Y_1kk_{4, 3}+xx_Y_1kk_{4,
     | xx_S_1xx_Fkk_{3, 4}-xx_Y_1kk_{4, 3}-xx_Y_1kk_{4, 5}                   
     | -xx_Exx_P_0kk_{6, 7}+xx_X_2kk_{7, 6}+xx_Y_2kk_{10, 11}                
     | xx_Exx_P_0kk_{6, 7}-xx_X_2kk_{7, 6}-xx_X_2kk_{7, 8}                   
     | -xx_Fxx_P_1kk_{9, 10}+xx_X_2kk_{7, 8}+xx_Y_2kk_{10, 9}                
     | xx_Fxx_P_1kk_{9, 10}-xx_Y_2kk_{10, 9}-xx_Y_2kk_{10, 11}               
     ------------------------------------------------------------------------
                                            |
     2}+xx_X_2kk_{7, 6}+xx_X_2kk_{7, 8}     |
                                            |
                                            |
      5}+xx_Y_2kk_{10, 9}+xx_Y_2kk_{10, 11} |
                                            |
                                            |
                                            |
                                            |
                                            |

             10      1
o6 : Matrix R   <-- R

Generate the steady-state equations in a specific ring.

i7 : N = twoLayerCascadeL()

o7 = S_0+E-->X_1
     X_1-->S_0+E
     X_1-->E+S_1
     S_1+P_0-->X_2
     X_2-->S_1+P_0
     X_2-->S_1+P_1
     E+P_0-->X_3
     X_3-->E+P_0
     X_3-->E+P_1
     S_1+F_1-->Y_1
     Y_1-->S_1+F_1
     Y_1-->S_0+F_1
     P_1+F_2-->Y_2
     Y_2-->P_1+F_2
     Y_2-->P_0+F_2

o7 : ReactionNetwork

i8 : R = createRing(N, ZZ/2)

o8 = R

o8 : PolynomialRing

i9 : F = steadyStateEquations(N, ZZ/2)

o9 = | xx_S_0xx_Ekk_{0, 1}+xx_X_1kk_{1, 0}+xx_Y_1kk_{10, 11}                
     | xx_S_0xx_Ekk_{0, 1}+xx_Exx_P_0kk_{6, 7}+xx_X_1kk_{1, 0}+xx_X_1kk_{1, 
     | xx_S_0xx_Ekk_{0, 1}+xx_X_1kk_{1, 0}+xx_X_1kk_{1, 2}                  
     | xx_S_1xx_P_0kk_{3, 4}+xx_S_1xx_F_1kk_{9, 10}+xx_X_1kk_{1, 2}+xx_X_2kk
     | xx_S_1xx_P_0kk_{3, 4}+xx_Exx_P_0kk_{6, 7}+xx_X_2kk_{4, 3}+xx_X_3kk_{7
     | xx_S_1xx_P_0kk_{3, 4}+xx_X_2kk_{4, 3}+xx_X_2kk_{4, 5}                
     | xx_P_1xx_F_2kk_{12, 13}+xx_X_2kk_{4, 5}+xx_X_3kk_{7, 8}+xx_Y_2kk_{13,
     | xx_Exx_P_0kk_{6, 7}+xx_X_3kk_{7, 6}+xx_X_3kk_{7, 8}                  
     | xx_S_1xx_F_1kk_{9, 10}+xx_Y_1kk_{10, 9}+xx_Y_1kk_{10, 11}            
     | xx_S_1xx_F_1kk_{9, 10}+xx_Y_1kk_{10, 9}+xx_Y_1kk_{10, 11}            
     | xx_P_1xx_F_2kk_{12, 13}+xx_Y_2kk_{13, 12}+xx_Y_2kk_{13, 14}          
     | xx_P_1xx_F_2kk_{12, 13}+xx_Y_2kk_{13, 12}+xx_Y_2kk_{13, 14}          
     ------------------------------------------------------------------------
                                              |
     2}+xx_X_3kk_{7, 6}+xx_X_3kk_{7, 8}       |
                                              |
     _{4, 3}+xx_X_2kk_{4, 5}+xx_Y_1kk_{10, 9} |
     , 6}+xx_Y_2kk_{13, 14}                   |
                                              |
      12}                                     |
                                              |
                                              |
                                              |
                                              |
                                              |

             12      1
o9 : Matrix R   <-- R

Substitute ReactionRates

The example below demonstrates how to substitute randomly generated values, for the reaction rates, directly into the steady state equations. For specific user-input values create a list with the corresponding values for each reaction rate; to view the order: N.ReactionRates, where N is the name of the reaction network.

i10 : M = reactionNetwork "A <--> 2B, A+C <--> D, D --> B+E, B+E --> A+C"

o10 = A-->2B
      2B-->A
      A+C-->D
      D-->A+C
      D-->B+E
      B+E-->A+C

o10 : ReactionNetwork

i11 : R = createRing M

o11 = R

o11 : PolynomialRing

i12 : K = toList(apply(0..length M.ReactionRates-1, i-> random(QQ)))

       9  1  9  1     3
o12 = {-, -, -, -, 1, -}
       2  2  4  2     4

o12 : List

i13 : Rr = toList(apply(0..length M.ReactionRates-1, i-> value(M.ReactionRates#i)))

o13 = {kk      , kk      , kk      , kk      , kk      , kk      }
         {0, 1}    {1, 0}    {2, 3}    {3, 2}    {3, 4}    {4, 2}

o13 : List

i14 : P = toList(apply(0..length Rr-1, i-> Rr#i=>sub(K#i,R)))

                   9              1              9              1          
o14 = {kk       => -, kk       => -, kk       => -, kk       => -, kk      
         {0, 1}    2    {1, 0}    2    {2, 3}    4    {3, 2}    2    {3, 4}
      -----------------------------------------------------------------------
                        3
      => 1, kk       => -}
              {4, 2}    4

o14 : List

i15 : SSE = flatten entries steadyStateEquations M

         2                                                                   
o15 = {xx kk       - xx xx kk       + xx xx kk       - xx kk       + xx kk   
         B  {1, 0}     A  C  {2, 3}     B  E  {4, 2}     A  {0, 1}     D  {3,
      -----------------------------------------------------------------------
               2
        , - 2xx kk       - xx xx kk       + 2xx kk       + xx kk      , -
      2}       B  {1, 0}     B  E  {4, 2}      A  {0, 1}     D  {3, 4}   
      -----------------------------------------------------------------------
      xx xx kk       + xx xx kk       + xx kk      , xx xx kk       -
        A  C  {2, 3}     B  E  {4, 2}     D  {3, 2}    A  C  {2, 3}  
      -----------------------------------------------------------------------
      xx kk       - xx kk      , - xx xx kk       + xx kk      }
        D  {3, 2}     D  {3, 4}      B  E  {4, 2}     D  {3, 4}

o15 : List

i16 : toList apply(0..length SSE-1, i-> sub(SSE#i,P))

       1  2   9         3         9      1         2   3                    
o16 = {-xx  - -xx xx  + -xx xx  - -xx  + -xx , - xx  - -xx xx  + 9xx  + xx ,
       2  B   4  A  C   4  B  E   2  A   2  D      B   4  B  E      A     D 
      -----------------------------------------------------------------------
        9         3         1     9         3       3
      - -xx xx  + -xx xx  + -xx , -xx xx  - -xx , - -xx xx  + xx }
        4  A  C   4  B  E   2  D  4  A  C   2  D    4  B  E     D

o16 : List

Ways to use steadyStateEquations :

steadyStateEquations(ReactionNetwork)
steadyStateEquations(ReactionNetwork,Ring)
steadyStateEquations(ReactionNetwork,InexactFieldFamily) (missing documentation)

For the programmer

The object steadyStateEquations is a method function.

steadyStateEquations -- creates steady-state equations of a reaction network

Description

See also

Ways to use steadyStateEquations :

For the programmer