How to solve a 13 set of equations and evaluate their change over time?

I need to solve these set of equations. I thought about using matrices but I could not figure out how use "syms" to add so many variables. Also, how can evaluate the ODE equations over time (tspan = [0:100]?

%K Constants 
k1 = 0.3;
k2 = 0.3;
k3 = 0.4;
k4 = 0.05;
k5 = 0.77;
k6 = 0.125;
k7 = 0.48;
k8 = 0.0017;
k9 = 0.27;
%Rate Equations
v1 = k1*ATP;
v2 = k2*PEP;
v3 = k3*P2G;
v4 = k4*BPG*ATP;
v5 = k5*G3P;
v6 = k6*FBP;
v7 = k7*DHAP*G3P;
v8 = k8*FBP;
v9 = k9*FBP;
v10 = K10*ADP;
%ODE
PEP = v1-v2;
P2G= v2-v3;
P3G = v3-v4;
BPG = v4-v5;
G3P = v5-v6-v7;
DHAP = v6-v7;
FBP = v7-v8-v9;
ATP = -v1-v4+v10;
ADP = v1+v4-v10;
%Here i try to do it for the PEP equation but i keep getting error. 
for tspan = 0:10
    v2 = k2*(v1-v2);
    dpdt = @(PEP,t) (k2*PEP*t);
    [PEP, t] = ode45(dpdt,tspan,0);
end

[Torsten]

As already noted, write out your equations in a mathematical notation since we cannot make sense of your code.

[John D'Errico]

What you have written does not make complete sense.

You define v2 outside the loop. Then inside the llop, you redefine v2 as

v2 = k2*(v1-v2)

But then you never use v2 anyway.

And your ODE?

dp/dt = k2*PEP*t

PEP is defined as

PEP = v1 - v2

But is that value of v2 changing? Do we know that?

What I'm not clear of, is this a DAE? So a differential algebraic system of equations?

You have 10 rate equations, for v1 through v10. Then you define 9 other algebraic equations, for the variables PEP, etc., as linear combinations of those rates.

So it LOOKs as it you have a DAE system. I'm confused as to what you are doing with v2 in the loop there.

Solving a DAE can be done using two numerical solvers.

help ode15s

 ODE15S Solve stiff differential equations and DAEs, variable order method.
    [TOUT,YOUT] = ODE15S(ODEFUN,TSPAN,Y0) with TSPAN = [T0 TFINAL] integrates 
    the system of differential equations y' = f(t,y) from time T0 to TFINAL 
    with initial conditions Y0. ODEFUN is a function handle. For a scalar T 
    and a vector Y, ODEFUN(T,Y) must return a column vector corresponding 
    to f(t,y). Each row in the solution array YOUT corresponds to a time
    returned in the column vector TOUT.  To obtain solutions at specific
    times T0,T1,...,TFINAL (all increasing or all decreasing), use TSPAN =
    [T0 T1 ... TFINAL].     
    
    [TOUT,YOUT] = ODE15S(ODEFUN,TSPAN,Y0,OPTIONS) solves as above with default
    integration properties replaced by values in OPTIONS, an argument created
    with the ODESET function. See ODESET for details. Commonly used options
    are scalar relative error tolerance 'RelTol' (1e-3 by default) and vector
    of absolute error tolerances 'AbsTol' (all components 1e-6 by default).  
    If certain components of the solution must be non-negative, use
    ODESET to set the 'NonNegative' property to the indices of these
    components.  The 'NonNegative' property is ignored for problems
    where there is a  mass matrix. 
    
    The Jacobian matrix df/dy is critical to reliability and efficiency. Use
    ODESET to set 'Jacobian' to a function handle FJAC if FJAC(T,Y) returns 
    the Jacobian df/dy or to the matrix df/dy if the Jacobian is constant. 
    If the 'Jacobian' option is not set (the default), df/dy is approximated 
    by finite differences. Set 'Vectorized' 'on' if the ODE function is coded 
    so that ODEFUN(T,[Y1 Y2 ...]) returns [ODEFUN(T,Y1) ODEFUN(T,Y2) ...]. 
    If df/dy is a sparse matrix, set 'JPattern' to the sparsity pattern of
    df/dy, i.e., a sparse matrix S with S(i,j) = 1 if component i of f(t,y)
    depends on component j of y, and 0 otherwise.    
 
    ODE15S can solve problems M(t,y)*y' = f(t,y) with mass matrix M(t,y). Use
    ODESET to set the 'Mass' property to a function handle MASS if MASS(T,Y) 
    returns the value of the mass matrix. If the mass matrix is constant, 
    the matrix can be used as the value of the 'Mass' option. Problems with
    state-dependent mass matrices are more difficult. If the mass matrix does
    not depend on the state variable Y and the function MASS is to be called
    with one input argument T, set 'MStateDependence' to 'none'. If the mass
    matrix depends weakly on Y, set 'MStateDependence' to 'weak' (the
    default) and otherwise, to 'strong'. In either case the function MASS is
    to be called with the two arguments (T,Y). If there are many differential
    equations, it is important to exploit sparsity: Return a sparse
    M(t,y). Either supply the sparsity pattern of df/dy using the 'JPattern'
    property or a sparse df/dy using the Jacobian property. For strongly
    state-dependent M(t,y), set 'MvPattern' to a sparse matrix S with S(i,j)
    = 1 if for any k, the (i,k) component of M(t,y) depends on component j of
    y, and 0 otherwise.    
 
    If the mass matrix is non-singular, the solution of the problem is
    straightforward. See examples FEM1ODE, FEM2ODE, BATONODE, or
    BURGERSODE. If M(t0,y0) is singular, the problem is a differential-
    algebraic equation (DAE). ODE15S solves DAEs of index 1. DAEs have
    solutions only when y0 is consistent, i.e., there is a yp0 such that
    M(t0,y0)*yp0 = f(t0,y0). Use ODESET to set 'MassSingular' to 'yes', 'no',
    or 'maybe'. The default of 'maybe' causes ODE15S to test whether M(t0,y0)
    is singular. You can provide yp0 as the value of the 'InitialSlope'
    property. The default is the zero vector. If y0 and yp0 are not
    consistent, ODE15S treats them as guesses, tries to compute consistent
    values close to the guesses, and then goes on to solve the problem. See
    examples HB1DAE or AMP1DAE.  
 
    [TOUT,YOUT,TE,YE,IE] = ODE15S(ODEFUN,TSPAN,Y0,OPTIONS) with the 'Events'
    property in OPTIONS set to a function handle EVENTS, solves as above 
    while also finding where functions of (T,Y), called event functions, 
    are zero. For each function you specify whether the integration is 
    to terminate at a zero and whether the direction of the zero crossing 
    matters. These are the three column vectors returned by EVENTS: 
    [VALUE,ISTERMINAL,DIRECTION] = EVENTS(T,Y). For the I-th event function: 
    VALUE(I) is the value of the function, ISTERMINAL(I)=1 if the integration 
    is to terminate at a zero of this event function and 0 otherwise. 
    DIRECTION(I)=0 if all zeros are to be computed (the default), +1 if only 
    zeros where the event function is increasing, and -1 if only zeros where 
    the event function is decreasing. Output TE is a column vector of times 
    at which events occur. Rows of YE are the corresponding solutions, and 
    indices in vector IE specify which event occurred.    
    
    SOL = ODE15S(ODEFUN,[T0 TFINAL],Y0...) returns a structure that can be
    used with DEVAL to evaluate the solution or its first derivative at 
    any point between T0 and TFINAL. The steps chosen by ODE15S are returned 
    in a row vector SOL.x.  For each I, the column SOL.y(:,I) contains 
    the solution at SOL.x(I). If events were detected, SOL.xe is a row vector 
    of points at which events occurred. Columns of SOL.ye are the corresponding 
    solutions, and indices in vector SOL.ie specify which event occurred. 
 
    Example
          [t,y]=ode15s(@vdp1000,[0 3000],[2 0]);   
          plot(t,y(:,1));
      solves the system y' = vdp1000(t,y), using the default relative error
      tolerance 1e-3 and the default absolute tolerance of 1e-6 for each
      component, and plots the first component of the solution.
 
    See also ODE23S, ODE23T, ODE23TB, ODE45, ODE23, ODE113, ODE15I,
             ODESET, ODEPLOT, ODEPHAS2, ODEPHAS3, ODEPRINT, DEVAL,
             ODEEXAMPLES, VDPODE, BRUSSODE, HB1DAE, FUNCTION_HANDLE.

    Documentation for ode15s
       doc ode15s

help ode23t

 ODE23T  Solve moderately stiff ODEs and DAEs, trapezoidal rule.
    [TOUT,YOUT] = ODE23T(ODEFUN,TSPAN,Y0) with TSPAN = [T0 TFINAL] integrates 
    the system of differential equations y' = f(t,y) from time T0 to TFINAL 
    with initial conditions Y0. ODEFUN is a function handle. For a scalar T 
    and a vector Y, ODEFUN(T,Y) must return a column vector corresponding 
    to f(t,y). Each row in the solution array YOUT corresponds to a time
    returned in the column vector TOUT.  To obtain solutions at specific
    times T0,T1,...,TFINAL (all increasing or all decreasing), use TSPAN =
    [T0 T1 ... TFINAL].     
    
    [TOUT,YOUT] = ODE23T(ODEFUN,TSPAN,Y0,OPTIONS) solves as above with default
    integration parameters replaced by values in OPTIONS, an argument created
    with the ODESET function. See ODESET for details. Commonly used options
    are scalar relative error tolerance 'RelTol' (1e-3 by default) and vector
    of absolute error tolerances 'AbsTol' (all components 1e-6 by default).  
    If certain components of the solution must be non-negative, use
    ODESET to set the 'NonNegative' property to the indices of these
    components.  The 'NonNegative' property is ignored for problems
    where there is a mass matrix. 
    
    The Jacobian matrix df/dy is critical to reliability and efficiency. Use
    ODESET to set 'Jacobian' to a function handle FJAC if FJAC(T,Y) returns 
    the Jacobian df/dy or to the matrix df/dy if the Jacobian is constant. 
    If the 'Jacobian' option is not set (the default), df/dy is approximated 
    by finite differences. Set 'Vectorized' 'on' if the ODE function is coded 
    so that ODEFUN(T,[Y1 Y2 ...]) returns [ODEFUN(T,Y1) ODEFUN(T,Y2) ...]. 
    If df/dy is a sparse matrix, set 'JPattern' to the sparsity pattern of
    df/dy, i.e., a sparse matrix S with S(i,j) = 1 if component i of f(t,y)
    depends on component j of y, and 0 otherwise.    
 
    ODE23T can solve problems M(t,y)*y' = f(t,y) with mass matrix M(t,y). Use
    ODESET to set the 'Mass' property to a function handle MASS if MASS(T,Y) 
    returns the value of the mass matrix. If the mass matrix is constant, 
    the matrix can be used as the value of the 'Mass' option. Problems with
    state-dependent mass matrices are more difficult. If the mass matrix does
    not depend on the state variable Y and the function MASS is to be called
    with one input argument T, set 'MStateDependence' to 'none'. If the mass
    matrix depends weakly on Y, set 'MStateDependence' to 'weak' (the
    default) and otherwise, to 'strong'. In either case the function MASS is
    to be called with the two arguments (T,Y). If there are many differential
    equations, it is important to exploit sparsity: Return a sparse
    M(t,y). Either supply the sparsity pattern of df/dy using the 'JPattern'
    property or a sparse df/dy using the Jacobian property. For strongly
    state-dependent M(t,y), set 'MvPattern' to a sparse matrix S with S(i,j)
    = 1 if for any k, the (i,k) component of M(t,y) depends on component j of
    y, and 0 otherwise.    
 
    If the mass matrix is non-singular, the solution of the problem is
    straightforward. See examples FEM1ODE, FEM2ODE, BATONODE, or
    BURGERSODE. If M(t0,y0) is singular, the problem is a differential-
    algebraic equation (DAE). ODE23T solves DAEs of index 1. DAEs have
    solutions only when y0 is consistent, i.e., there is a yp0 such that
    M(t0,y0)*yp0 = f(t0,y0). Use ODESET to set 'MassSingular' to 'yes', 'no',
    or 'maybe'. The default of 'maybe' causes ODE23T to test whether M(t0,y0)
    is singular. You can provide yp0 as the value of the 'InitialSlope'
    property. The default is the zero vector. If y0 and yp0 are not
    consistent, ODE23T treats them as guesses, tries to compute consistent
    values close to the guesses, and then goes on to solve the problem. See
    examples HB1DAE or AMP1DAE.    
 
    [TOUT,YOUT,TE,YE,IE] = ODE23T(ODEFUN,TSPAN,Y0,OPTIONS) with the 'Events'
    property in OPTIONS set to a function handle EVENTS, solves as above 
    while also finding where functions of (T,Y), called event functions, 
    are zero. For each function you specify whether the integration is 
    to terminate at a zero and whether the direction of the zero crossing 
    matters. These are the three column vectors returned by EVENTS: 
    [VALUE,ISTERMINAL,DIRECTION] = EVENTS(T,Y). For the I-th event function: 
    VALUE(I) is the value of the function, ISTERMINAL(I)=1 if the integration 
    is to terminate at a zero of this event function and 0 otherwise. 
    DIRECTION(I)=0 if all zeros are to be computed (the default), +1 if only 
    zeros where the event function is increasing, and -1 if only zeros where 
    the event function is decreasing. Output TE is a column vector of times 
    at which events occur. Rows of YE are the corresponding solutions, and 
    indices in vector IE specify which event occurred.    
 
    SOL = ODE23T(ODEFUN,[T0 TFINAL],Y0...) returns a structure that can be
    used with DEVAL to evaluate the solution or its first derivative at 
    any point between T0 and TFINAL. The steps chosen by ODE23T are returned 
    in a row vector SOL.x.  For each I, the column SOL.y(:,I) contains 
    the solution at SOL.x(I). If events were detected, SOL.xe is a row vector 
    of points at which events occurred. Columns of SOL.ye are the corresponding 
    solutions, and indices in vector SOL.ie specify which event occurred. 
 
    Example
          [t,y]=ode23t(@vdp1000,[0 3000],[2 0]);   
          plot(t,y(:,1));
      solves the system y' = vdp1000(t,y), using the default relative error
      tolerance 1e-3 and the default absolute tolerance of 1e-6 for each
      component, and plots the first component of the solution.
 
    See also ODE15S, ODE23S, ODE23TB, ODE45, ODE23, ODE113, ODE15I,
             ODESET, ODEPLOT, ODEPHAS2, ODEPHAS3, ODEPRINT, DEVAL,
             ODEEXAMPLES, VDPODE, BRUSSODE, HB1DAE, FUNCTION_HANDLE.

    Documentation for ode23t
       doc ode23t