function [t,y]=ASModel(tspan,y0,theta,kx,ky,r0,s0,a,zeta,lambda,l,w)
% This function solves the dynamical system for modelling the paddling of
% two flagella on the same cell, the model is the anisotropic spring model.
%
%             y = [phi_1,phi_2,r1,r2] = solution
%             tspan = the time interval to be solved
%             y0=[phi_1;phi_2;r1;r2] = initial condition at t0
%             theta = the angle between the motor in the cell and the flagella bead
%             kx = spring constant along x direction
%             ky = spring constant along y direction
%             r0 = equilibrium radius for radial spring r
%             s0 = equilibrium radius for radial spring s
%             a = radius of the flagella bead
%             lambda = [r-spring const, s-spring const, eta-spring const]
%             l = distance between effective center of flagella
%             w = [w1, w2]=intrinsic angular frequency
% Parameter switch_s is used to include or exclude the s spring
%==========================================================================
switch_s=0; % 1 if spring s exists, 0 if s is rigid
Atol=1e-13;Rtol=1e-11;  % absolute and relative tolerance for solver

kr=lambda(1); ks=lambda(2); eta=lambda(3);
w1=w(1); w2=w(2);
options=odeset('AbsTol',Atol,'RelTol',Rtol);
[t,y]=ode45(@derivative,tspan,y0,options);
switch switch_s
    case 0
        s1_=ones(length(t),1)*s0;
        s2_=s1_;
    case 1
        Q_=@(a,b)(kx-ky)/2*cos(a+b+2*theta)+(kx+ky)/2*cos(a-b);
        U12_=-Q_(y(:,1),y(:,2)-2*theta)./(ks+eta+Q_(y(:,1),y(:,1)));
        U21_=-Q_(y(:,2)-2*theta,y(:,1))./(ks+eta+Q_(y(:,2)-2*theta,y(:,2)-2*theta));
        E12_=s0*(ks+eta)./(ks+eta+Q_(y(:,1),y(:,1)))-eta*(y(:,3)-r0)./(ks+eta+Q_(y(:,1),y(:,1)));
        E21_=s0*(ks+eta)./(ks+eta+Q_(y(:,2)-2*theta,y(:,2)-2*theta))-eta*(y(:,4)-r0)./(ks+eta+Q_(y(:,2)-2*theta,y(:,2)-2*theta));
        
        s1_=-(E12_ - E21_.*U12_)./(U12_.*U21_ - 1);
        s2_=-(E21_ - E12_.*U21_)./(U12_.*U21_ - 1);
end
y=[y s1_ s2_];

    function dydt=derivative(t,y)
        % solve for explicit expression for each derivative symbolically
        %        A=[  r1  X12    0  Y12;
        %            X21   r2  Y21    0;
        %              0  L12    1  T12;
        %            L21    0  T21    1;];
        %        b=[ I12;
        %            I21;
        %            O12;
        %            O21];
        %        x=A\b;
        %==================================================================
        % Define functions: sin, cos, N, M, A, B, G, Q used in Equations
        c=@(a,b) cos(y(a)-y(b));
        s=@(a,b) sin(y(a)-y(b));
        m=@(a,b,n) (-1)^(n-1)*l*sin(y(a))+y(2+b)*sin(y(a)-y(b));
        n=@(a,b,n) (-1)^n*l*cos(y(a))+y(2+a)-y(2+b)*cos(y(a)-y(b));
        A=@(a,b,n) (-1)^(n-1)*l*y(2+b)*sin(y(b))+y(2+a)*y(2+b)*sin(y(a)-y(b));
        B=@(a,b,n) (-1)^n*l*cos(y(b))+y(2+a)*cos(y(a)-y(b))-y(2+b); 
        G=@(a,b)(ky-kx)/2*sin(a+b+2*theta)-(ky+kx)/2*sin(a-b);
        Q=@(a,b)(kx-ky)/2*cos(a+b+2*theta)+(kx+ky)/2*cos(a-b);
        %==================================================================
        % Put in labels for 1 and 2
        K=3/4*a; r1=y(3); r2=y(4);
        c12=c(1,2);
        c21=c(2,1);
        s12=s(1,2);
        s21=s(2,1);
        g12=G(y(1),y(2)-2*theta);
        g21=G(y(2)-2*theta,y(1));
        g11=G(y(1),y(1));
        g22=G(y(2)-2*theta,y(2)-2*theta);
        r12=sqrt(l^2-2*l*(r1*cos(y(1))-r2*cos(y(2)))+r1^2+r2^2-2*r1*r2*c12);
        m121=m(1,2,1); % M_{12}^+
        m212=m(2,1,2); % M_{21}^-
        n121=n(1,2,1);
        n212=n(2,1,2);
        a121=A(1,2,1);
        a212=A(2,1,2);
        b121=B(1,2,1);
        b212=B(2,1,2);
        %==================================================================
        switch switch_s
            case 0
                s1=s0;
                s2=s0;
            case 1
                % Solve expression for s1,s2 explicitly
                %      A=[   1 U12;
                %          U21   1];
                %      b=[ E12;
                %          E21];
                %      x=A\b;
                U12=-Q(y(1),y(2)-2*theta)/(ks+eta+Q(y(1),y(1)));
                U21=-Q(y(2)-2*theta,y(1))/(ks+eta+Q(y(2)-2*theta,y(2)-2*theta));
                E12=s0*(ks+eta)/(ks+eta+Q(y(1),y(1)))-eta*(r1-r0)/(ks+eta+Q(y(1),y(1)));
                E21=s0*(ks+eta)/(ks+eta+Q(y(2)-2*theta,y(2)-2*theta))-eta*(r2-r0)/(ks+eta+Q(y(2)-2*theta,y(2)-2*theta));
                
                s1=-(E12 - E21*U12)/(U12*U21 - 1);
                s2=-(E21 - E12*U21)/(U12*U21 - 1);
        end
        
        % Define elements for matrix using coefficients from equations
        X12=-K/r12*(m121*a121/r12^2+r2*c12);
        Y12=-K/r12*(m121*b121/r12^2+s21);
        X21=-K/r12*(m212*a212/r12^2+r1*c21);
        Y21=-K/r12*(m212*b212/r12^2+s12);
        L12=-K/r12*(n121*a121/r12^2+r2*s12);
        T12=-K/r12*(n121*b121/r12^2+c12);
        L21=-K/r12*(n212*a212/r12^2+r1*s21);
        T21=-K/r12*(n212*b212/r12^2+c21);
        I12=r0*w1+s1*s2/(zeta*r1)*g12-s1^2/(zeta*r1)*g11;
        I21=r0*w2+s1*s2/(zeta*r2)*g21-s2^2/(zeta*r2)*g22;
        O12=-(kr+eta)/zeta*(r1-r0)-eta/zeta*(s1-s0);
        O21=-(kr+eta)/zeta*(r2-r0)-eta/zeta*(s2-s0);
        
        
        dydt =[
            
        (I21*X12 - I12*r2 + I12*L12*Y21 - O12*X12*Y21 + O21*Y12*r2 + I21*L12*T21*Y12 - I21*T12*T21*X12 - L12*O21*Y12*Y21 + O21*T12*X12*Y21 + I12*T12*T21*r2 - O12*T21*Y12*r2)/(X12*X21 - r1*r2 + L12*Y21*r1 + L21*Y12*r2 - L12*L21*Y12*Y21 + L12*T21*X21*Y12 + L21*T12*X12*Y21 - T12*T21*X12*X21 + T12*T21*r1*r2)
        (I12*X21 - I21*r1 + I21*L21*Y12 - O21*X21*Y12 + O12*Y21*r1 + I12*L21*T12*Y21 - I12*T12*T21*X21 - L21*O12*Y12*Y21 + O12*T21*X21*Y12 + I21*T12*T21*r1 - O21*T12*Y21*r1)/(X12*X21 - r1*r2 + L12*Y21*r1 + L21*Y12*r2 - L12*L21*Y12*Y21 + L12*T21*X21*Y12 + L21*T12*X12*Y21 - T12*T21*X12*X21 + T12*T21*r1*r2)
        (O12*X12*X21 - I12*L12*X21 + I21*L12*r1 - O12*r1*r2 - I21*L12*L21*Y12 + I21*L21*T12*X12 + L12*O21*X21*Y12 - O21*T12*X12*X21 - I12*L21*T12*r2 + L21*O12*Y12*r2 + O21*T12*r1*r2)/(X12*X21 - r1*r2 + L12*Y21*r1 + L21*Y12*r2 - L12*L21*Y12*Y21 + L12*T21*X21*Y12 + L21*T12*X12*Y21 - T12*T21*X12*X21 + T12*T21*r1*r2)
        (O21*X12*X21 - I21*L21*X12 + I12*L21*r2 - O21*r1*r2 - I12*L12*L21*Y21 + I12*L12*T21*X21 + L21*O12*X12*Y21 - O12*T21*X12*X21 - I21*L12*T21*r1 + L12*O21*Y21*r1 + O12*T21*r1*r2)/(X12*X21 - r1*r2 + L12*Y21*r1 + L21*Y12*r2 - L12*L21*Y12*Y21 + L12*T21*X21*Y12 + L21*T12*X12*Y21 - T12*T21*X12*X21 + T12*T21*r1*r2)];
    end

end