This document discusses using MATLAB to simulate control of magnetized Tetrahymena pyriformis cells. It provides code for an ODE simulation that models how cells' orientations change over time due to a rotating magnetic field, causing their trajectories to follow spiral patterns. The code defines parameters like frequency and velocity, initializes cell positions and orientations, runs an ODE solver to simulate changes over time, and plots the resulting cell paths.

1. COMPUTATIONAL BIOLOGY USING MATLAB
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SIMULATE CONTROL OF MAGNETIZED TETRAHYMENA PYRIFORMIS CELLS
This sample assignment shows uniform magnetic field uses control-Lypunov function to steer all cells
to orbit goal positions
ODEsimSpirographPlot
function ODEsimSpirographPlot
% makes a pretty plot of many robots being driven by a constantly rotating
% magnetic field.
global freq vel a
set(0,'defaultaxesfontsize',14);
set(0,'defaulttextfontsize',14);
set(0,'DefaultTextInterpreter', 'latex')
format compact
clc
freq = 10; % as the frequency increases, the lines become straighter, but the differentiation
decreases)
vel = 0.7; %mm/s (Estimate by Yan Ou) 700 mum/s
tF = 10; %final time
%initial conditions for cells (x,t,theta)
[y,x] = meshgrid(1:2,1:4);
x = reshape(x,numel(x),1);
y = reshape(y,numel(y),1);
N = numel(x); %number of robots
a = linspace(4,11,N)'; % a values, should be between 3 and 10
th = 0*ones(N,1); % performance is very dependent on initial orientation for large amplitude
options = odeset('RelTol',1e-6,'AbsTol',1e-6 );
[T,Y] = ode45(@simPyriformis,[0 tF],[x;y;th],options);
%%%%%%%%%%%%%%%%%%%%%%% PLOTTING
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%draw path, cells

2. figure(1)
clf
c = distinguishable_colors(N);
for i = 1:N
s = plot(Y(1,i),Y(1,i+N),'x','color',c(i,:));
uistack(s,'bottom');
hold on
p = plot(Y(:,i),Y(:,i+N),'-','color',c(i,:));
uistack(p,'bottom');
text(Y(1,i),Y(1,i+N)-.2,['$a$ = ',num2str(a(i))],'HorizontalAlignment','center','Interpreter','latex')
end
axis equal
title({[num2str(N),' cells simulated for ',num2str(tF),' s'];},'Interpreter','latex');%['theta_M =
',num2str(Mag),'sin(',num2str(freq),'t)']});
xlabel('$x$ mm','Interpreter','latex')
ylabel('$y$ mm','Interpreter','latex')
for i = 1:N
drawCell(Y(end,i),Y(end,i+N),Y(end,i+2*N),a(i),0.05);
end
%fix the axis and xticks
set(gca, 'xtick',[-5:10],'ytick',[-5:10])
axis([-.5,4.5,.5,2.8])
%saveas(gcf,'../../pictures/pdf/SpiralsFromRotatingMagneticField.pdf')
format_ticks(gca,{'0','1','2','3','4'},{'1','2'},[0:4],[1:2],0,0,0.01);
xlabel({' ';'$x$ mm'},'Interpreter','latex')
ylabel({'$y$ mm';' '},'Interpreter','latex')
set(gcf,'PaperUnits','inches')
set(gcf,'papersize',[7,4])
set(gcf,'paperposition',[-.5,-.5,8,5])
print -dpdf '../../pictures/pdf/SpiralsFromRotatingMagneticField.pdf'
end
function dq = simPyriformis(t,q)
% simulates a differential equation model for the cells
% q = [x_coordinates; y_coordinates; theta_coordinates,
% magnetic_field_orientation]
global freq vel a
dq = zeros(size(q)); % a column vector for the change in state
N = (numel(q))/3; %number of robots
th = q(2*N+1:3*N); %current orientations of the cells

3. dq(1:N) = vel*cos(th);
dq(N+1:2*N) = vel*sin(th);
angDif = t*freq-th;
dq(2*N+1:3*N) = a.*sin(angDif);
end
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