Mathematical Models and Assumptions
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A mass-balance-based quantitative mathematical model was developed by Huang et al. to
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evaluate the probability of segregational plasmid loss in both batch suspended and biofilm
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growth cultures (Huang et al. 1993). Here, we describe a new approach, using a mathematical
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model combined with reporter-gene technology, in which the number of plasmid-bearing and
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plasmid-free cells can be enumerated by detecting different fluorescent colors expressed by
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various reporter genes encoded on either plasmids or chromosomal DNA. The utility of this
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technology allows for the non-disruptive in situ observation of pDNA fate, such as localization
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of the donor cells as well as the plasmid distribution within biofilms without affecting cell
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viability in the biofilm itself.
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I.
Liquid Culture Model
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This model is based on a mass balance between plasmid-bearing (X+) and plasmid-free (X-
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) cells in a suspended liquid culture that can be described by two processes: cell growth and
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growth-related plasmid segregational loss.
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dX 
   X   p  X 
dt
dX 
   X   p  X 
dt
(1)
(2)
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Where p is the probability of plasmid loss, µ+ and µ- are specific growth rates of plasmid-
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bearing and plasmid-free cells (1/time). Let χ = X-/X+ and applying the product rule of
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differentiation, after transformation of the equations, yields,
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dX 
d
dX 
X

dt
dt
dt
(3)
Plotting (1) and (2) into (3), yields,
d
 (       p  )   p 
dt
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(4)
Assuming     m and     m , where  m and  m are maximum specific growth rate of
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plasmid-bearing and plasmid-free cells (time-1). Equation (1) and (4) can be written as:
dX 


 (m
 p m
)X 
dt
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d
 (  m   m  p m )   p m
dt
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(5)
(6)
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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
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parameters can be solved as:  m = m+ + b;  m = m+ + m, and p = b/ (m+ + b).
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II.
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Biofilm Culture Model
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Three processes are involved in net accumulation of a biofilm cell culture: (1) the
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cellular deposition of planktonic cells from the liquid phase; (2) adherent cell growth within a
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matrix of extracellular polymers on the substratum; and (3) biofilm-bound cell detachment from
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the surface (Bryers 2000). In the biofilm culture mathematical model, deposition of suspended
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cells is neglected because of the experimental design in which suspended cells are not provided
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to the flow cell after the inoculation process. Since bacterial residence time in the flow cell was
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maintained ≤ 0.70 minutes, which is much less than the generation time of the culture (~ 90
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minutes), one can safely assume that any cells leaving the flow cell are bacteria that originated
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as debris from the biofilm.
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Assuming plasmid-bearing and plasmid-free cells have identical constant rates of
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adhesion and detachment under the same hydrodynamics conditions, mass balances for plasmid-
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bearing (B+) and plasmid-free (B-) cell in a biofilm culture can be derived that account for: cell
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growth, growth-related plasmid loss, and biofilm cell detachment.
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dB
   B   p  B   kd Bd
dt
(10)
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dB
   B   p  B   kd Bd
dt
(11)
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Where B+ and B- are the densities of plasmid-bearing and plasmid-free cells, respectively,
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in the biofilm; p is the probability of plasmid loss, µ+ and µ- are the specific growth rates of
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plasmid-bearing and plasmid-free cells, respectively, in a biofilm (1/time); Bd and Bd are the
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densities of detached plasmid-bearing and plasmid-free cells, respectively, at the outer regions
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of biofilms (cell/L2); and Kd is the detachment rate constant (1/t).
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Assuming both plasmid-bearing and plasmid-free cells detach in the same ratio from the
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biofilms. As the overall ratio in the bioiflms, let β = B-/B+ = B d / B d , applying rule of
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differentiation, and yields,
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
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)
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The specific growth rates for the two populations can be expressed as:

S
and    m S
  m
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Equation (13) can be written as
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
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
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This mathematical model shows that the probability of plasmid loss is only a function of
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the relative maximum growth rates of each population in the biofilm. Since growth rate affects
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p , then it is intuitive that limiting substrate concentrations could indirectly influence plasmid
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loss.
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III.
Assumptions
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Three assumptions were made in the original development of the mathematical model.
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First, deposition of suspended cells is ignored because experimentally, once the surface
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inoculation was complete, suspended bacterial cells were not supplied to the reactor. Second,
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during biofilm formation, the residence time in the flow cell reactor was maintained much lower
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than the generation time of the strain, essentially eliminating suspended cell replication in the
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fluid phase. Thus, combined with assumption one, any suspended cells leaving the reactor
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effluent must originate only from the biofilm due to the detachment process. Third, the model
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also assumes that these re-suspended cells generated as detached debris from the biofilm do not
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re-deposit onto the biofilm prior to leaving the flow cell.
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The validity of each assumption is addressed in the following paragraphs. Assumption
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One is valid based directly on experimentally maintaining a sterile inlet feed. Assumption Two
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is also valid based on experimentally maintaining the residence time in the flow cell reactor
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  V / F = 0.694 minutes (41.6 seconds), which is 130 times faster than the time require for P.
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putida to divide (~90.0 min).
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To estimate the validity of Assumption Three, one can estimate the maximum particle
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flux of cells to the flow cell surface. Worst case scenario in these experiments would be at the
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highest suspended cell concentrations leaving the flow cell, which has been estimated to be
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~1.0X106 cells/cm2. The equation by Bowen et al (1976) was used to estimate particle flux, Nx,
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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)
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where Cx is the average cell concentration in the fluid phase (1.0 x 106 cells/cm3), Dc is the
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diffusivity of the cells (~3.0 x 10-5 cm2/sec), h is half the distance between the two parallel
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plates (cm), and K is defined as:
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 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,
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 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
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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
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19,000 cells/cm2 falling onto 4.56 cm2 of biofilm. A actual concentration of biofilm at this
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ambient suspended cell concentration was 1.0x109 cells/cm2.
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So, at these conditions if re-deposition occurred as described in these calculations, then a
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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
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cm2 = 4.56x109 cells, thus contributing less than 0.0019% of the biofilm’s upper cell density.
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References:
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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.
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