Cell Division

Having established a very simple 2D bioreactor model in the previous section with a rough approximation to hydrodynamics and cell illumination history, this section focusses on providing a more accurate cell division system. The combination of the Photosynthetic Factory Model (PSF) [57,58] with the Kawasaki site-exchange model is presented and discussed in this section.

A thorough discussion of the PSF model is out scope. However, a brief summary is provided for self-containment. The PSF model facilitates a simplification of the photosynthetic process by considering it as a Markov process [57]. Three discrete states are allowed: activated, resting and inhibited. Cells may transition between these states with certain probabilities. The state diagram is shown in Figure7.7. Modifications added are simply the addition of cell division and increment states. The former allows the cell to split depending onP(split), and the latter increments a counter. The counter is used to influence the growth rate, and is described later.

State transitions depend on the probabilities specified in the diagram shown. Parameters were hand-selected for

Pα,Pβ,Pγ andPδ(the symbols in the state diagram are subscripted to avoid conflicting use) to0.7,0.5,0.3and

0.4respectively. A cell may divide if it is in the resting state, and depending onP(split), which is discussed below. In order to observe the effects of photo-inhibition and limitation, the counter (now the activation state counter) is modified to measure how long a cell spends in the activated state. This is then used to add a small contribution on to the cell division probability,P(split), which is also discussed below.

158 7. PHOTOBIOREACTOR MODELLING

Cell Division Open (Resting)

State Activated State

Inhibited State _{Increment Count}

δ

βI

αI

γ

P(split)

FIGURE7.7: State diagram of the Photosynthetic Factory model and additional flow.

N=L2

for square lattice

blank lattice, initialise 10 random cells for alltime-stepsdo

for all3x3 blocks in latticedo

for all9 sitesiin each block, random orderdo choose a random neighbour sitej

compute energy change ifi, jexchanged ifenergy fallsthen

accept change and do exchange else

compute Metropolis probabilityp obtain random probabilityr1

accept change conditionally onr1< p end if

ifin activated statethen

increment activated state counter end if

ifin resting statethen

compute cell division probabilityP(split)

obtain random probabilityr2 divide onr2< P(split) end if state transition end for end for end for

7.2. CONVENTIONAL ABM 159

ComputingP(split)is modified to include a contribution from the activated state counter. The effect of light intensity decay is still included. The new formula for computing this quantity is shown in Equation7.3. This equation shows the probability of a cell dividing wheref is the distance to the nearest wall of the reactor vessel. In this equation,βis set to a constant15, unless noted otherwise. The second and third terms in the equation are the effects of the activation state counters. This is added to ensure that the time a cell spends in the activated state corresponds to a slightly higher cell division rate [211]. The curve this produces againstais logarithmic with a

y-intercept of zero.

P(split) =γe−βf2−ln( a

40+ 2)

−1_{+ (ln 2)}−1 _(7.3)

An additional modification from the initial design presented in the previous section is that the medium is now inoculated by choosing a set of random sites. This method attempts to mimic the effect of injecting a sample of a strain of algae into a well-mixed medium. The system state after such a sample inoculation some 100 timesteps fromt= 0is shown in Figure7.8.

FIGURE7.8: Model configuration showing photo stimulated agent preferential growth at right and left of simulated bioreactor.

A plot of the activation state counters are shown in Figure7.9. The plot shows the highest and lowest counter, as well as a standard deviation and mean. The data shown has been averaged across 100 independent runs.

Modifying the light decay parameterγhas a dramatic effect on the fill rate of the bioreactor. Several curves for different values ofγare shown in Fig.7.10. Each curve was averaged over 100 independent runs. The horizontal line through the curves at the centre indicate the point (t1/2) the half-life of the system is considered to have been reached. The quantityt1/2is defined here as the number of time steps taken to reach a fill fraction of 50% in the bioreactor.

As can be seen from the fill fractions (growth curves), the maximum of 100% fill fraction is reached logarithmi- cally. In practice, bioreactors are harvested after having reached a suitable culture density, which could perhaps be described as a fill fraction from 40% to 60%, considering that the lattice the simulation is built on is discrete. It may therefore be useful to compute the time that is likely needed for the reactor to fill to a certain level. This “half-life” of the systemt1/2can be determined from the data collected.

160 7. PHOTOBIOREACTOR MODELLING

FIGURE7.9: Plot of the activation state counters by time step. The data shown is averaged over 100 independent runs.

7.2. CONVENTIONAL ABM 161

Plotting the light intensity parameter valuesγagainstt1/2obtained in the system yields the plot shown in Figure7.11. This plot suggests a relationship betweent1/2andγshown in Equation7.4. Each point in this plot is averaged over 100 separate runs, and the error bars represent the standard deviations of the points.

FIGURE7.11: Log-log Plot of the brightness scale parameter (γ) against the time steps taken to reach a fill fraction of0.5(t1/2).

t1/2≈ec·γm (7.4)

The negative slope in the plot shown in Figure7.11suggestsm=−0.26±0.02andc = 5.34±0.09. As expected, increasing illumination intensity while keeping the exponential light decay in the medium can only reduce

t1/2to a certain degree. The effects of mutual cell shading is still not considered, but will be discussed in the next section.

The light decay parameterβvariation in Figure7.12appears to show a very weak linear relationship witht1/2. Such a relationship reaffirms that lower values ofβ(corresponding to a more clear liquid with less light decay) improves growth rates. In practice, mutual cell shading is the major contributor in controlled environments to light decay in the medium. The use of the Lambert-Beer law for light decay in the medium in this sense, was simply approximating this mutual cell shading while crudely assuming that all cells are distributed equally throughout the medium.

It would appear that the amalgamation of the PSF model with the Kawasaki site spin exchange model provides loosely realistic growth kinetics in the simulation of a photobioreactor. However, negative cell growth is not accounted for (cell deaths), and also mutual cell shading is disregarded but approximated assuming that culture

162 7. PHOTOBIOREACTOR MODELLING

FIGURE7.12: Plot of the (β) exponent against the time taken to reacht=t1/2 = 0.5. Here, each point is averaged over 100 independent runs, and error bars represent standard deviations. Theγvalue for these runs was0.032.

7.2. CONVENTIONAL ABM 163

density is uniform across the entire reactor. In the next section, these two assumptions are eliminated, and the simulation is expanded to 3 dimensions, so as to mimic the geometry of a flat panel photobioreactor.