Cell Shape Model

Background

One of our research themes focuses on the ability of L-form bacteria to fit the mould and/or fill spaces by growing into them. We would like to explore the biophysical properties of the cell and the limits of the cell size the L-form can grow to before dividing by constructing a 2-dimentional cell model.

The shape of the bacterial cell is usually determined by the structures of the cell wall and bacterial cytoskeleton. The cell wall of B.subtilis is a thick layer of peptidoglycan which contains cell-wall proteins, techoic and lipotechoic acids. Peptidoglycan is a rigid structure which helps to protect the cell from osmotic pressure and mechanical damage. It also helps the cell to maintain a constant shape, while not specifically defining it. This role is played by the cytoskeleton. The bacterial cytoskeleton is largely similar to that of eukaryotes. It is comprised of three types of protein structures: FtsZ, MreB and crescentin (only present in certain species) which are homologous to eukaryotic tubulin, actin and intermediate filaments respectively. These proteins also perform other functions in the cell; usually they are involved in cell growth and division and therefore are often essential for the cell's survival. The communication between the two regulatory components is assured by a series of membrane-associated enzymes (penicillin binding proteins). (Cabeen, Jacobs-Wagner, 2005)

Because the L-forms lack the peptidoglycan cell wall, cells naturally adopt the most energetically favourable shape (i.e. sphere) regardless of the cytoskeleton structure. With protoplasts (cells with chemically digested cell wall), the growth, variation in shape and division is restricted due to the limited amount of the cell membrane which simply ruptures if the pressure inside the cell increases. The L-forms created with our bio-brick will have a mutation in the MurE gene which causes spontaneous increase in production of the IspA gene product which results in increased synthesis of the cell membrane. This stabilises the cell and allows for growth and division.

Approach

For the purposes of the study the complex model of the growing cell inside of the confined space can be broken down to simpler models of the system at three phases. The first phase would be a constantly growing cell, followed by a model of the cell when it touches the boundaries and therefore changes the trajectory of the membrane growth and starts adopting the shape of the boundaries.

Unlimited Cell Growth Model

Physiological basis

Since the L-Forms do not have the cell wall, the shape of the cell is determined by the laws of Newtonian physics and therefore will be as close to the shape of a sphere as possible. For the purposes of this model we will assume that the cell is spherical to start with and therefore will expand evenly in all directions. L-Forms of Bacillus subtilis made with our BioBrick will have an induced mutation in the ispA gene, which allows the cell to constitutively produce cell membrane phospholipids and therefore the surface area of the membrane will increase over time as the cell grows. Provided that the cell has a sufficient nutrient supply, phospholipids will be produced at a constant rate p, and transported to the surface of the membrane at rate T. Once at the surface, they are inserted into the membrane at rate i, which is determined by the maximum velocity of an enzyme x. These processes will result in an increase of the sphere’s radius at a rate dR(t)/dt=f(R,T,i,p). In the paper written by D. Fanelli and A.J. McKane on "Thermodynamics of vesicle growth and instability" it is explained that the growth of a vesicle is quasistatic in nature, so an equation for growth of the radius is given:

 R(t)=R(0)e(λt)/2, where λ=ln 2/Td

However, this model seems physiologically unrealistic. As to allow for the exponential growth of the radius, the cell would have to produce lipids at an exponentially increasing rate, which is larger than the rate of increase of the radius by a factor of 2π. This seems highly unrealistic, as even at infinite concentration of source lipids inside the cell, the rate at which they are converted to phospholipids and transported to the membrane must be limited by the concentration of enzymes and their turnover rate, which is expected to be quasistatic to support homeostasis.

Mathematical basis

Limitation of Growth by a Set Boundary

Model

Unlimited Cell Growth

MATLAB code

%plots a time dependent circle with a growing radius which stops growing

	cellv=10^(-15);
	t_end=10;
	m=100;
	lambda=10^(-10);
	M0=10^(-17);
	K=10^(-10);
	a=10^(-4);
	n=100;
	tspan = linspace(0,t_end,m);
	
	r0=(cellv/pi*3/4)^(1/3);
	alpha=(3*lambda)/(32*pi^2);
	
	x=zeros(n);
	y=zeros(n);
	
	x=linspace(-a,+a,n);
	y=linspace(-a,+a,n);
	
	figure(1)
	clf
	axis([-a +a -a +a])
	hold on
	
	r=r0;
	k=0;

	while r<=a
		
		k=k+1;
		t=tspan(k);
		r=(5*alpha*(M0*t+0.5*K*t^2)+r0^5)^(1/5);
		c(k)= 2*pi*r;

                rectangle('Position',[-r,-r,r*2,2r], 'Curvature',[1], 'Edgecolor',[0,0,1], Facecolour, [0,0,1])

		plot(x,y)
		pause(0.5)
		radius(k)=r;

	end;

        hold all
        l=0;
        circ=c(k);
        crv=1

        while crv>0.1
              
		k=k+1;
		t=tspan(k);
		r=(5*alpha*(M0*t+0.5*K*t^2)+r0^5)^(1/5);
		cn(k)= 2*pi*r;

                l=((cn(k)-circ/4);
                crv=1-2*l/a;

                rectangle('Position',[-r,-r,r*2,2r], 'Curvature',[crv], 'Edgecolor',[0,0,1], Facecolour, [0,0,1])

		plot(x,y)
		pause(0.5)
		radius(k)=r;

       end;

Deformation of the Cell

Simulation