bit24953-sm-0001-SuppData

advertisement
Mathematical Models and Assumptions
1
2
A mass-balance-based quantitative mathematical model was developed by Huang et al. to
3
evaluate the probability of segregational plasmid loss in both batch suspended and biofilm
4
growth cultures (Huang et al. 1993). Here, we describe a new approach, using a mathematical
5
model combined with reporter-gene technology, in which the number of plasmid-bearing and
6
plasmid-free cells can be enumerated by detecting different fluorescent colors expressed by
7
various reporter genes encoded on either plasmids or chromosomal DNA. The utility of this
8
technology allows for the non-disruptive in situ observation of pDNA fate, such as localization
9
of the donor cells as well as the plasmid distribution within biofilms without affecting cell
10
viability in the biofilm itself.
11
I.
Liquid Culture Model
12
This model is based on a mass balance between plasmid-bearing (X+) and plasmid-free (X-
13
) cells in a suspended liquid culture that can be described by two processes: cell growth and
14
growth-related plasmid segregational loss.
15
16
dX 
   X   p  X 
dt
dX 
   X   p  X 
dt
(1)
(2)
17
Where p is the probability of plasmid loss, µ+ and µ- are specific growth rates of plasmid-
18
bearing and plasmid-free cells (1/time). Let χ = X-/X+ and applying the product rule of
19
differentiation, after transformation of the equations, yields,
20
21
22
dX 
d
dX 
X

dt
dt
dt
(3)
Plotting (1) and (2) into (3), yields,
d
 (       p  )   p 
dt
1
(4)
Assuming     m and     m , where  m and  m are maximum specific growth rate of
1
2
plasmid-bearing and plasmid-free cells (time-1). Equation (1) and (4) can be written as:
dX 


 (m
 p m
)X 
dt
3
d
 (  m   m  p m )   p m
dt
4
5
6
7
(5)
(6)
11
Plotting dX+/dt vs. X+. According to Equation (5), the slope is:
m+ =  m  p m
(7)
d
Plotting
vs.  . According to Eqn (6), respectively, the slope and intercept are:
dt
slope =  m   m  p m
(8)
intercept = p m
(9)
Since there are three equations of (7), (8) and (9) and the three unknowns, three unknown
12
parameters can be solved as:  m = m+ + b;  m = m+ + m, and p = b/ (m+ + b).
13
II.
8
9
10
Biofilm Culture Model
14
Three processes are involved in net accumulation of a biofilm cell culture: (1) the
15
cellular deposition of planktonic cells from the liquid phase; (2) adherent cell growth within a
16
matrix of extracellular polymers on the substratum; and (3) biofilm-bound cell detachment from
17
the surface (Bryers 2000). In the biofilm culture mathematical model, deposition of suspended
18
cells is neglected because of the experimental design in which suspended cells are not provided
19
to the flow cell after the inoculation process. Since bacterial residence time in the flow cell was
20
maintained ≤ 0.70 minutes, which is much less than the generation time of the culture (~ 90
21
minutes), one can safely assume that any cells leaving the flow cell are bacteria that originated
22
as debris from the biofilm.
23
Assuming plasmid-bearing and plasmid-free cells have identical constant rates of
24
adhesion and detachment under the same hydrodynamics conditions, mass balances for plasmid-
2
1
bearing (B+) and plasmid-free (B-) cell in a biofilm culture can be derived that account for: cell
2
growth, growth-related plasmid loss, and biofilm cell detachment.
3
dB
   B   p  B   kd Bd
dt
(10)
4
dB
   B   p  B   kd Bd
dt
(11)
5
Where B+ and B- are the densities of plasmid-bearing and plasmid-free cells, respectively,
6
in the biofilm; p is the probability of plasmid loss, µ+ and µ- are the specific growth rates of
7
plasmid-bearing and plasmid-free cells, respectively, in a biofilm (1/time); Bd and Bd are the
8
densities of detached plasmid-bearing and plasmid-free cells, respectively, at the outer regions
9
of biofilms (cell/L2); and Kd is the detachment rate constant (1/t).
10
Assuming both plasmid-bearing and plasmid-free cells detach in the same ratio from the
11
biofilms. As the overall ratio in the bioiflms, let β = B-/B+ = B d / B d , applying rule of
12
differentiation, and yields,
13
14
15

dB 
d
dB 
 B

dt
dt
dt
(12)
Substituting equation (4.12) and (4.13) into (4.14), and yields,
d
 (       p  )   p 
dt
(13)
17
The specific growth rates for the two populations can be expressed as:

S
and    m S
  m
18
Equation (13) can be written as
16
19
20
21
22
23
24
25

Ks  S
Ks  S
d
S
S
S
S
 ( m  m  p m )  p m
dt
Ks  S Ks  S
Ks  S
Ks  S
(14)
Plot dβ/dt versus β, according to equation (14), the slope and intercept will be:

m’ = (m  m  pm ) m S
(15)
Ks  S
b’= p  S

m
(16)
Ks  S
The probability of plasmid loss within biofilm, p , can be solved from equation (15) and (16) as
follows:
b' m  m
(17)
p
m'b'
m
3
1
This mathematical model shows that the probability of plasmid loss is only a function of
2
the relative maximum growth rates of each population in the biofilm. Since growth rate affects
3
p , then it is intuitive that limiting substrate concentrations could indirectly influence plasmid
4
loss.
5
III.
Assumptions
6
Three assumptions were made in the original development of the mathematical model.
7
First, deposition of suspended cells is ignored because experimentally, once the surface
8
inoculation was complete, suspended bacterial cells were not supplied to the reactor. Second,
9
during biofilm formation, the residence time in the flow cell reactor was maintained much lower
10
than the generation time of the strain, essentially eliminating suspended cell replication in the
11
fluid phase. Thus, combined with assumption one, any suspended cells leaving the reactor
12
effluent must originate only from the biofilm due to the detachment process. Third, the model
13
also assumes that these re-suspended cells generated as detached debris from the biofilm do not
14
re-deposit onto the biofilm prior to leaving the flow cell.
15
The validity of each assumption is addressed in the following paragraphs. Assumption
16
One is valid based directly on experimentally maintaining a sterile inlet feed. Assumption Two
17
is also valid based on experimentally maintaining the residence time in the flow cell reactor
18
  V / F = 0.694 minutes (41.6 seconds), which is 130 times faster than the time require for P.
19
putida to divide (~90.0 min).
20
To estimate the validity of Assumption Three, one can estimate the maximum particle
21
flux of cells to the flow cell surface. Worst case scenario in these experiments would be at the
22
highest suspended cell concentrations leaving the flow cell, which has been estimated to be
4
1
~1.0X106 cells/cm2. The equation by Bowen et al (1976) was used to estimate particle flux, Nx,
2
to a surface in a rectangular parallel flow cell (Bowen 1976),
  2 1/3 


C D  9K 
  
x
c
N 
x
h  0.8934 




3
(18)
4
5
where Cx is the average cell concentration in the fluid phase (1.0 x 106 cells/cm3), Dc is the
6
diffusivity of the cells (~3.0 x 10-5 cm2/sec), h is half the distance between the two parallel
7
plates (cm), and K is defined as:
8
9
10
 D c  8L 
 
(19)
4v
h
3h


 ave 
where vave is the average velocity in flow cells (L/t), and L is the length of the flow cell (cm).
K  
For the conditions used in these experiments,
11
12
13
14
 3.0x105 cm2 /sec  83.8cm 

K= 
 = 5.6x10-4 • 81.1 = 4.54 x10-2
 40.107cm/sec0.125cm 30.125cm


And, thus, Nx would be
16
1/3

 
2


 
 1.0x10 6 cells/cm3 (3.0x10 05 cm2 /sec)   
9 4.5x10 2  

2



Nx

 = 457 cells/cm -sec.
0.125cm
0.8934

 





With a 41.6 second residence time in the flow cell, this flux would translate to a total of
17
19,000 cells/cm2 falling onto 4.56 cm2 of biofilm. A actual concentration of biofilm at this
18
ambient suspended cell concentration was 1.0x109 cells/cm2.
15
5
1
So, at these conditions if re-deposition occurred as described in these calculations, then a
2
total of 19000 cells/cm2 x 4.56 cm2 = 8.66x104 cells would fall onto a bed of 1.0 x 109 x 4.56
3
cm2 = 4.56x109 cells, thus contributing less than 0.0019% of the biofilm’s upper cell density.
6
1
References:
2
3
4
5
6
7
Bowen BD, S. Levine, and N. Epstein. 1976. Fine Particle Deposition in Laminar Flow through
Parallel-Plate and Cylindrical Channels. J . Colloid Interface Sci. 54:375.
Bryers JD. 2000. Biofilms - II. J. Wiley-liss Publishers, New York. NY.
Huang CT, Peretti SW, Bryers JD. 1993. Plasmid retention and gene expression in suspended
and biofilm cultures of recombinant Escherichia coli DH5-pMJR1750). Biotechnol
Bioeng 41(2):211-20.
7
Download