Download Predicting Individual Bacterium Cell Growth Behavior from

Predicting Individual Bacterium Cell Growth Behavior from Population Information
Abstract
We explore the kinetics of bacterial cells to predict the dependency of growth and division rates on cell length and
age. Our model is applied to the analysis of two bacterial populations, a wild-type strain of Bacillus subtilis, and a
minicell-producing strain that carries the divID B1 mutation. The results show that our model describes the wild-type
population very well. In the case of the minicell-producing mutant we find evidence that the rate of cell division must
be a function not only of cell size but also of cell age.
1
Introduction
Bacteria only have a single, circular double-stranded DNA molecule, and hence a single chromosome. Bacterial have
the apparatus to grow, duplicate their genome, and divide quickly. Bacteria multiply by elongating, forming a septum
at mid-cell, and finally completing the septum and dividing into two cells. This asexual method of reproduction is
called binary fission. One basic question is to understand how growth and division depend on the cell’s length and
age. This care and time necessary to develop an experiment and witness these processes has been a major deterrent.
An alternative approach is to develop a mathematical model that depends on some measurable parameters to predict
how growth and division rates age governed by cell length and age.
2
Mathematical Model
Collins and Richmond suggested that the necessary relationships can be found in measuring cell lengths of a bacterial
population during its “log-phase”. They postulated that the growth rate of an individual cell depends only on x, the
length of the cell.
ẋ = V (x).
(1)
Focusing on the properties of the population, they introduce three probability density functions • ψ(x) - for the daughter cells at division,
Figure 1: Rod-Shaped Bacterium, hemorrhagic E. coli, strain 0157:H7 (division) (SEM x22,810).
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Statistics Case Studies: Bacterial Growth Behavior
Math 363 (Spring 2009)
Figure 2: Binary fission of bacteria
• φ(x) - for the mother cell at division, and
• λ(x) - the steady-state length density function.
- and a number k, the instantaneous growth rate of the population.
The Collins and Richmond equation establishes the relationship of these functions
1 d
(V (x)λ(x)) = 2φ(x) − φ(x) − λ(x).
k dx
Generally speaking, k and λ can easily be determined in the laboratory. V can be determined with some care. φ and
ψ are difficult.
In our model, we keep the Collins and Richmond strategy to model grown. To describe cell division, we introduce
a intensity function ν that depends on both the length of the cell and its age a.
ν(x, a) ≈ Px,a {division in the time interval [t, t + ∆t]}.
In addition, we investigate the relationship between age and length as a determining factor in cell division. In our
study, we consider both a wild-type strain of the soil bacteria Bacillus subtilis and a mutant strain. In the mutant strain,
a fraction p of the cell divisions do not take place at the center of the cell but rather near one of the poles. This type
of cell division produces on nucleated normal cell and on anucleated “minicell”. For normal cell division, we assume
that cells divide exactly and consequently ψ(x) = 2φ(2x). The governing equation is
∂
∂
λ(x, a) +
(V (x)λ(x, a))
∂a
∂x
= − (k + ν(x, a)) λ(x, a),
(2)
with
Z
∞
λ(x, 0) = 4(1 − p)
ν(2x, a)λ(2x, a) da
Z ∞ 0
+p
ν(x + α, a)λ(x + α, a) da,
(3)
0
where α is the length of a minicell.
The left side of equation (2) tells us that a colony of cells ages at a constant rate but grows at the rate given in
(1). The right side indicates that these changes are balanced by the overall growth of the colony and by cell division.
When a cell divides, the resulting daughter cells have age a = 0. Equation (4) says that new cells are born with length
x from mother cells of length 2x and some age a from normal cells and length x + α for minicells. The first integral
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Statistics Case Studies: Bacterial Growth Behavior
Math 363 (Spring 2009)
Figure 3: Colony of Paenibacillus vortex bacteria
gives the contribution over all ages for normal cell division, then second for minicell division. The constant k is not a
parameter in the model, but rather is chosen so that the λ remains a probability density.
To compare equation (), we remove the dependence on age and set the fraction of minicell divisions at p = 0. this
yields
∂
(V (x)λ(x)) = − (k + ν(x)) λ(x),
(4)
∂x
3
Material and Methods
To test the model, we take data from the steady-state length distribution of two populations of Bacillus subtilis 168
- a wild-type strain 1085 and a mutant strain, RHX that carries a defect in the DivIV-B1 gene. Cultures were grown
on complex agar medium (TBAB). Fluid cultures (20ml) of the same composition (TB) were initiated by toothpick
transfer from fresh colonies. Populations were allowed to grow at 23◦ C with 25 rpm shaking until it reached mid-log
phase, usually between 8 and 10 hours after innoculation. Samples were taken and transferred to glass slides, allowed
to dry, and heat fixed to the slides by warming from below. Cells were stained with crystal violet and viewed with
a Wild M20 phase-constast microscope. Images were obtained with a Cohu CCD video camera and printed with a
Mitsubishi Video Graphics Processor. Cell lengths were measured from printed images and converted to true length
using the image of a stage micrometer scale.
4
Results
To test our model, we begin with the wild-type. First, we assume that wild-type cells do not grow beyond a certain
length xmax . Thus, V (x) = 0 for x ≥ xmax . Second, we assume that wild-type cells do not divide below a certain
length xmin . The mathematically simplest model consistent with these assumption are for the cell growth rate
γ(xmax − x)(x − b), xmin /2 ≤ x ≤ xmax ,
V (x) =
(5)
0
otherwise.
and cell division intensity
ν(x) =
β, x ≥ 2xmin ,
0 xmin ≤ x < 2xmin .
(6)
for constants β, γ and b ≤ xmin .
The choice of a quadratic function for V is consistent with the findings of Harvey. For cell division, we find no
cells smaller than a given minimum size. Also, Lasota and Mackey have observed that the difference in division time
between sister cells has an exponential distribution. The choice of ν(x) = β for x ≥ xmin is consistent with this
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Statistics Case Studies: Bacterial Growth Behavior
Math 363 (Spring 2009)
Figure 4: Experimental (histogram) and theoretical (smooth curve) length distribution Bacillus subtilis strain 1085.
observation. A choice of a discontinuous ν is unrealistic but reflects the fact that the change in division rate is very
rapid compared to the time-scale of cell growth.
Multiplication of β and γ by a constant c has the effect of running the process by a factor c faster. Because we are
collecting data in steady state, this change will not affect the steady-stante distribution λ. As a consequence, our model
has four parameters, xmax , xmin , b, and β/γ. The parameter values that best fit the experimental data B. subtilis are
xmin = 3.25, b = 0.25, β/γ = 24. The resulting graph of λ is shown with the experimental data in Figure 4. The
solution selected a stead state growth rate k = 1.76. The chi2 goodness of fit test on the data yields a P -value of 0.57.
Figure 5: Minicell mutant B. subtilis RHX. Beyond producing minicells, this mutant produces a linear clone of unseparated cells.
The purpose in studying the mutant strain is to try to distinguish the importance of cell age and cell length as
factors in determining the timing of cell division. These two attributes are highly correlated in normal cell division,
On the other hand, the products of a minicell producing fission is an anucleated minicell that is quite short and a
nucleated sister cell that is quite long. As can me seen in Figure the minicell producing mutant has a dramatically
different distribution of cell lengths.
The minicell model has a additional parameter p, the fraction of cell divisions that result in a minicell. In the
mutant strain for this study, the cells do not separate upon septation. Thus, we estimate p by examining many clones
and reporting the empirical value p̂ = 0.38.
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Statistics Case Studies: Bacterial Growth Behavior
Math 363 (Spring 2009)
Figure 6: Histogram showing both lengths of both normal strain and minicell producing strain.
To take into account length in cell division intensity we choose:
β, x ≥ 2xmin , and a ≥ amin .
ν(x, a) =
0 otherwise.
(7)
In order to solve the pair of equations (2) and (4), we simulate the process of growth and division to estimate
λ(x, 0). We then use the method of characteristics to to solve equation (2) and check the accuracy of the solution
using equation (4). The integration of the function λ(x, a) with respect to the age a gives the steady-state length
distribution. The solution for the four model situations are displayed in Figure (7).
The inclusion of age in the wild type results in a small increase in the peak of the density function. This shows
how similar the information from age and length are for wild-type bacteria. If we attempt to model minicell length
distribution with no dependence on age, we obtain a small broadening of the range of cell lengths. Finally, when both
age and length are included, the shape of the density shifts dramatically giving a excellent fit to the data.
5
Discussion
Using a relatively simple data on steady state length distribution, this modeling strategy provides some insights into
the kinetics of bacterial growth. Using this approach, we are able to describe details of individual cell behavior using
population level information. For example, the integration of the function λ(x, a) with respect to the length x gives the
steady-state age distribution. Such a distribution would be extremely tedious to obtain in the laboratory and as Figure
(8) shows, it is relatively insensitive to the choice of model.
The next stage for the modeling of bacterial cell division is to recognize that cell length and age is serving as a
surrogate for the genetic and biochemical pathways that regulate cell growth and septum formation. Thus, to make
progress, we must move from models based on the morphological kinetics of the cell to one of it biochemical dynamics.
p
0
0
0.38
0.38
amin
0
0.75
0
0.75
wild-type model
normal cells - age a factor
minicells - age not a factor
minicells - age a factor
Table I: Modeling parameters
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Statistics Case Studies: Bacterial Growth Behavior
Math 363 (Spring 2009)
Figure 7: The four predicted length distribution and the experimental distribution for the minicell mutant strain.
Figure 8: The four predicted steady state age distributions.
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Statistics Case Studies: Bacterial Growth Behavior
Math 363 (Spring 2009)
References
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219- 230.
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agilis. J. Bacteriol. 93, 605-617.
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[7] Koch, A. L. and Schaechter, M. (1962). A model for statistics of the cell division process. J. Gen. Microbiol. 29, 435- 454.
[8] Koch, A. L. and Holtje, J.-V. (1995). A physical basis for the precise location of the division site of rod-shaped bacteria: the
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[9] Koch, A. L. (1995). Bacterial Growth and Form. New York, NY: Chapman & Hall.
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[12] Marr, A. G., Harvey, R. J. and Trentini, W. C. (1966) Growth and division of Escherichia coli. J. Bacteriol. 91, 2388.
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Statistics Case Studies: Bacterial Growth Behavior
Math 363 (Spring 2009)
STATISTICAL PROCEDURES
The main statistical procedures here is to find the theoretical stationary distribution that best matches the data.
covariance. We shall accomplish this by taking values of the stationary distribution at 1 micron intervals and comparing
these to the data using the χ2 distribution.
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