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Modeling Heterogeneous Bacterial Populations Exposed to
Antibiotics: The Logistic-Dynamics Case
Pratik R. Bhagunde1, Vincent H. Tam2, and Michael Nikolaou1,3
1Chemical & Biomolecular Engineering Department, University of Houston
2Department of Clinical Sciences and Administration, College of Pharmacy, University of Houston
3Author to whom all correspondence should be addressed. [email protected]
Abstract. In typical in vitrotests for clinical use or development of antibiotics, samples from abacterial population are exposed to the antibiotic at various concentrations. The resulting data
are then used to build a mathematical model suitable for dosing regimen design or for further
development. For bacterial populations that include resistant subpopulations an issue that
has reached alarming proportions building such a model is challenging. In prior work we
developed a related modeling framework for such heterogeneous bacterial populations
following linear dynamics when exposed to an antibiotic. We extend this framework to the
case of logistic dynamics, common among strongly resistant bacterial strains. Explicit formulas
are developed that can be easily used in parameter estimation and subsequent dosing regimen
design under realistic pharmacokinetic conditions. A case study using experimental data fromthe effect of an antibiotic on a gram-negative bacterial population exemplifies the usefulness of
the proposed approach.
KEYWORDS: Antibiotic resistance, nonlinear dynamics, logistic growth, bacterial population,
modeling, cumulant
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1 Introduction
The emergence of bacterial strains that resist attacks by antibiotics was observed as early as a
decade after the initial widespread use of penicillin1. By now, bacterial resistance to antibiotics
has reached such alarming proportions2-8
that a task-force was formed in 2014 by US
Government executive order to coordinate efforts for addressing this critical public-health
issue9.
Effective clinical use or development of antibiotics typically requires a series of tests,
one of which involves in vitroexposure of samples from a bacterial population to an antibiotic
at various concentrations (time-kill test). From the outcomes of such tests conclusions are
drawn on how to design effective antibiotic dosing regimens to accommodate the antibiotics
pharmacokinetics in vivo. A mathematical model is often employed to assist the process by
capturing the dynamics of the antibiotics bactericidal activity.
Because a bacterial population may contain bacteria that cannot be killed by anantibiotic, it is common to model the population as two subpopulations, one resistant and one
susceptible to the antibiotic.10,11
Experimental evidence for the presence of resistant bacteria
in a population is easily provided in a time-kill test when, for some antibiotic concentrations,
the bacterial population initially declines, due to killing of susceptible bacteria, and
subsequently starts growing, owing to resistant bacteria taking over.
While this resistant/susceptible modeling dichotomy is conceptually valuable, it may
yield models that provide misleading quantitative information regarding what antibiotic
concentrations would eradicate an entire bacterial population (by preventing eventual growth
of its most resistant subpopulation).12-17 In previous work17 we ascribed that modelingdeficiency to the fact that the bacterial rate of killing by the antibiotic (i.e. susceptibility or
resistance) can take a multitude of values spread over the bacterial population, rather than two
distinct values corresponding to resistant or susceptible subpopulations. Correspondingly, we
developed a mathematical modeling approach that accounts for this fact. In particular, we
showed that the proposed approach can make useful long-term predictions in cases where the
two-subpopulation approach fails.
The modeling approach in our previous work relied on the assumption of linear
dynamics for physiological growth of a bacterial population. As we explain in section 2, this
assumption is reasonable for a bacterial population in decline caused by exposure to an
antibiotic. However, the assumption is not accurate when a population is not in decline,
particularly not far from the populations saturation point. Such a situation may easily arise in
practice, when an antibiotic shows weak bactericidal activity against strongly resistant strains.
In such a situation, full nonlinear description of the bacterial population logisticdynamics not
far from population saturation would offer better accuracy. The objective of this paper is to
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develop the corresponding modeling approach. Specifically, the main idea is to develop for the
logistic-dynamics case the counterpart of results developed earlier for the linear-dynamics case.
While pursuing the main idea, we present newly discovered results that bear significance for
the linear case as well.
In the rest of the paper, background and motivation for this work is presented in moredetail in section2. The main theoretical results are presented in section3. A case study that
illustrates the usefulness of the formulas developed based on laboratory experimental data is
presented in section4,followed by discussion in section5. All proofs and experimental details
are presented in appendices.
2 Background and Motivation
As mentioned above, standard in vitro time-kill experiments expose samples of a bacterial
population to an antibiotic at various concentrations (usually twofold or fourfold dilutions) that
remain constant over time. Starting with a population of 0N similar bacterial cells in an
environment of an antibiotic at a time-invariant concentration C, a standard population
balance assuming logisticphysiological growth yields
max kill ratedue to antibiotic
logistic physiological growth rate
( )( ) 1 ( ) ( )
=
((
((((((g
dN N t K N t r C N t
dt N, 0(0)N N= (1)
where the bacterial population size ( )N t is treated as a continuous variable; gK is the specific
growth rate of the bacterial population; maxN is the saturation size of a bacterial population, at
which further growth stops; and ( )r C is the specific bacterial kill rate due to an antibiotic at
concentration C, assumed to be the same for all bacterial cells. A typical form of ( )r C is the
Hill expression10,18,19
50
( )H
k
H H
K Cr C
C C=
+ (2)
where kK is the asymptotic maximum value of the specific kill rate as C ; 50C is the
antibiotic concentration at which the kill rate reaches 50% of its asymptotic maximum value;
and H is the Hill exponent20
, which determines how sigmoidal (inflected) the shape of ( )r C is.
It is clear that for a population size far from its saturation point (i.e. max( )N t N
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populations are not homogeneous, namely they consist of subpopulations that experience
different killing rate constants ( )r C . (The growth rate constant gK is assumed to be the same
for all subpopulations, due to common physiology among bacteria of the same species, barring
small fluctuations due to bio-fitness cost incurred by resistant strains.) Consequently, eqn. (1)
is no longer valid and must somehow be modified, to account for the distribution of ( )r C overthe bacterial population.
Starting with eqn. (1) for max( )N t N
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3 Results
3.1 General case
Consider a heterogeneous bacterial population for which the bacterial kill rate induced by an
antibiotic at time-invariant concentration Cvaries among bacteria, with more resistant (less
susceptible) bacteria corresponding to lower values of the kill rate constant ( )r C . This setting
is plausible from a physiological viewpoint, given that variability is expected within any given
bacterial species, whether such variability is mild or pronounced. Therefore, the entire
bacterial population can be split into a number of subpopulations indexed by k , each
subpopulation having size ( )kN t , with
( ) ( )kk
N t N t= . (3)
Each subpopulation of size ( )kN t satisfies the counterpart of eqn. (1), namely
( )max
( )1 ( )k g k k
dN N tK r C N t
dt N
=
, 1,2,3,...k= . (4)
Note that the term max1 ( )N t N is common for all subpopulations, because it refers to
competition among bacteria for common resources. Note also that it is important to
In contrast to eqn. (1), of Bernoulli type with the standard closed-form solution
( )0
( )0 0
max max
( )1
( ) ( )e 1g
g
r C K t
g
r C
K
N t N N Nr C
N K N
= + +
, (5)
the set of equations in eqn. (4) does not admit a standard solution for ( )N t . Furthermore,
measuring each subpopulation size ( )kN t is practically infeasible; rather, only measurements
of ( )N t can reasonably be obtained. Therefore, a formula is needed for ( )N t as a function of
time for model parameter estimation. In addition, it is of paramount importance to estimate
the kill rate ( )kr C of the most resistant subpopulation, to ensure that an antibiotic
concentration Cis identified that guarantees eradication of the entire population, including itsmost resistant subpopulation under in vitro or in vivo pharmacokinetic conditions. It should
also be noted that the exact distribution of ( )kr C over the population is of secondary
importance.
Explicit formulas for ( )N t and estimate of ( )kr C for the most resistant subpopulation
are provided next. Note that index keventually drops out from all results.
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Theorem 1 Aggregate dynamics of heterogeneous bacterial population with logistic growth
Assume that a heterogeneous population of 0N bacteria is exposed to an antibiotic at time-
invariant concentration C. The population has a number of subpopulations, each of which
satisfies eqn. (4). Then the aggregate growth or decline of the entire bacterial population is
characterized by the following equations:
max
( )1 ( ) ( )g
dN N t K t N t
dt Nm
=
(6)
2( )d
tdt
m= (7)
2
3( )d
tdt
= (8)
1( )n
nd t
dt += , 3n (9)
where
( )( ) ( )
( )
kk
k
N tt r C
N tm = (10)
and
[ ]22 ( )( ) ( ) ( )
( )
kk
k
N tt r C t
N t m= (11)
are the average and variance, respectively, of the kill rate constant over the entire population
at time t; and ( )n t are the cumulants22, p. 928
of the kill rate constant distribution function
( ) ( )
( ), ( )
kk
N tf r C t
N t= (12)
at time t, defined through the moment- and cumulant-generating functions, ( , )M s t and
( , )s t , respectively, as follows.
( )( )( , ) ( ), ksr C kk
M s t e f r C t= . (13)
[ ] 1!1
( , ) ln ( , ) ( ) n
nn
n
s t M s t t s
=
= = . (14)
0
( , )( )
n
n n
s
s tt
s
=
=
, 1,2,...n= (15)
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with 1( ) ( )t t m= and2
2 ( ) ( )t t = .
Proof: SeeAppendix B.
Remark 1 Comparison of linear and logistic growth dynamics
Interestingly enough, the equations resulting from Theorem 1, proved for logistic-growth
dynamics, are remarkably similar to these in Theorem 1 of Nikolaou & Tam17
. In fact, eqns. (7)-
(9) turn out to be similar, whereas eqn. (6) is of the same nature, i.e. it is the logistic-growth
dynamics counterpart of the linear-growth dynamics equation
( )( ) ( )gdN
K t N t dt
m= (16)
Consequently, the following remarks can be made, which are the counterparts of the linear-
growth dynamics case.
- The first four cumulants ( )n t , 1,...,4n= , are directly related to the average, ,m
variance, 2 , skewness, 33m
, and kurtosis excess, 44 3
m
, of the kill rate constant
distribution, namely
1 m= ,2
2 = , 3 3 m= , and4
4 4 3 m = (17)
where ( )tm
are the central moments defined as
( ) ( )( ) ( ) ( ),k kk
t r t f r C t m m=
, 0 (18)
for the kill rate constant distribution function f .
- Because 2( ) 0t , eqn. (7) implies that the average kill rate constant ( )tm (a measure
of susceptibility or inverse measure of resistance) of a heterogeneous bacterial
population will decrease monotonically with time. Furthermore, because ( ) 0tm ,
( )tm will converge to a non-negative value. This is because more susceptible bacteria
will be preferentially eradicated by the antibiotic over more resistant bacteria, until only
the most resistant bacteria, if any, remain in the population.
Theorem 1 can be used to derive closed-form expressions for the time-dependence of the
bacterial population size as well as of the average and variance of the kill rate constant
distribution, as indicated next.
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Theorem 2 Closed-form expressions for time-dependence of population characteristics
Under the assumptions ofTheorem 1,the total bacterial population, kill rate constant average,
and kill rate constant variance depend explicitly on the moment- and cumulant-generating
functions of the initial distribution of the kill rate constant ( )r C as
0
00 max
( )ln ( , 0) ln 1 e ( , 0)g
t K
g g
NN tK t t K M d
N N
tt t
= + +
, (19)
2 2 31 13 42! 3!
( ,0)( ) (0) (0) (0) (0) ...
s t
st t t t
sm m
=
= = + +
, (20)
22 2 2 31 1
3 4 52! 3!2
( ,0)( ) (0) (0) (0) (0) ...
s t
st t t t
s
=
= = + +
, (21)
respectively. Moreover, the higher-order cumulants satisfy
2 31 11 2 32! 3!
( ,0)( ) (0) (0) (0) (0) ...
n
n n n n nn
s t
st t t t
s + + +
=
= = + +
, 3n (22)
Proof: SeeAppendix C.
Remark 2 Implications ofTheorem 2
- For a population size far from its saturation point, maxN , corresponding to
approximately linear-growth kinetics, eqn. (19) generalizes the result obtained by
Nikolaou & Tam17
as
0
( )ln ( , 0)g
N tK t t
N
= +
(23)
- The terms ( , 0)t and ( , 0)M t of eqn. (19), defined in eqns. (13) and (14),
respectively, refer to the cumulant-generating function and to the moment-generating
function of the initial bacterial population (they depend on the initial values of the
cumulants (0)n ).
- While the initial values (0)n of the cumulants may be practically impossible to
estimate from experimental data alone, it is demonstrated in the following section that
approximations of the initial distribution of the kill rate constant, ( )r C , over the
bacterial population results in convenient expressions that explicitly contain only a few
parameters, which can be estimated from standard experimental data, namely from the
size of the bacterial population at a few distinct points in time, measured during time-
kill experiments.
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3.2 Explicit formulas for specific distributions of the kill rate constant
It is interesting to examine the following two special cases of the above results, namely when
the initial ( )r C approximately follows the normal or Poisson distribution, because closed-form
expressions can be derived.
3.2.1 Normally distributed initial kill rate constant
Strictly speaking, it is impossible to have a distribution of the initial kill rate constant ( )r C that
is exactly normal, because that would entail negative values for ( )r C . Nevertheless, it is
interesting to consider an approximatelynormally distributed ( )r C , because, excluding the first
two cumulants (average and variance), all higher-order cumulants of the normal distribution
are identically22
(0) 0n = , 3n . (24)
In that case, the following explicit formulas can be developed with applicability over limited
time t.
Theorem 3 Explicit formulas for normally distributed initial kill rate constant
For initially normally distributed ( )r C with average m and variance 2 , it follows that
( ) ( )2 2
0
2 2 2 2max
2 212
0
( ) (0)
2 2 2 2
( )ln ( )
ln 1 exp erfi ,g g g
g
K K K t N
g N
N tK t t
N
K m m m p
m
+
= +
+
(25)
where erfi( ) erf( )z iz i= ,2
2
0erf( )
ztz e dt
p
= ,
2( )t tm m = (26)
and
2 2( )t = (27)
for a time period not exceeding
max 2t
m
= . (28)
In addition, for the same time period,
( ) 0n t = , 3n (29)
i.e. the distribution of ( )r C will remain normal for the same time period.
Proof: SeeAppendix D.
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The above normality assumption yields simple explicit formulas and provides reasonable
predictions for relatively short time periods, as has been demonstrated for linear growth
dynamics by Nikolaou & Tam17
. However, the normal distribution assumption is problematic
when predictions over longer periods of time are sought, e.g. because ( )tm , eqn. (26), becomes
negative. A better alternative is presented next, using a Poisson distribution.
3.2.2 Poisson-distributed initial kill rate constant
Assuming an underlying Poisson distribution for the kill rate constant ( )r C over the initial
bacterial population yields explicit formulas that are valid over arbitrarily long time periods and,
because they are explicit, they are convenient for parameter identification from total bacterial
population time-kill data alone. Specifically, the initial kill rate distribution function is assumed
to be
( ) min( )
( ),0 [ ]
!
k
k
r C r ef r C P k
a kX
= = =
((
, 0,1,2,...k= (30)
where the average and variance, 0> , of the Poisson-distributed variable X are related to
the average, m, and variance, 2 , of the kill rate constant, min( ) r C aX r = + , as
min 0a rm = + . (31)
2 2
min( ) 0a a r m= = . (32)
The parameter 0> roughly determines the shape of the discrete unimodal
distribution f , as can be visualized inFigure 2.
Figure 2. Family of the discrete distributions min !( ) [ ] ke
kf r P r r ak
= = + = , 0k , for the kill
rate constant ( )r C parametrized in terms of and minr . All distributions correspond to the
same average 1m= and 1a= . Note that ( ) 0f r = for 0r< .
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Given 0> , the parameters 2 , eqn. (32) and m , eqn. (31), determine the spread and
location of f , respectively. Equivalently, the parameters min 0r and 0a> in eqn. (30) are the
corresponding parameters for translation and dilation of ( )r C , respectively.
Within the above framework, the following explicit formulas can be developed.
Theorem 4 Explicit formulas for Poisson-distributed initial kill rate constant
Given eqn. (30), eqns. (19), (20), and (21) yield
( ) ( )
( ) ( )
( ) ( )
( )min
min
min
0
0min
max 0
min
0
max
( )ln 1
ln 1 exp 1
1
ln 1 , ,g
g
at
g
t
a
g g
at
g
K rr Kg ata
a
N tK r t e
N
NK K r e d
N
K r t e
K Ne e
a N
t
t t
= +
+ +
= +
+
(33)
( ) minmin min
min
( ) exp
at
rt r r t
r ae
mm m
= +
= +
(34)
and
( )2
min2 min
2
( ) exp
at
r rt t
a e
m m
=
=
(35)
respectively, where the parameters min ,r ,a ,m andminr
a
m
= depend on C , and the
incomplete gamma function is defined as
( ) 1
0
1
0 1, , z
c z
zc z z z e dz
= . (36)
Proof: SeeAppendix E.
Remark 3 Implications ofTheorem 4
- The entire bacterial population will be eradicated if and only if min gr K> .
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- The entire population will eventually be dominated by the most resistant subpopulation,
as eqn. (34) implies that minlim ( )t
t rm
= .
- The logarithmic part on the right-hand side of eqn. (33) is negligible when the
population size ( )N t is far from maxN , but becomes significant when maxN is close to
maxN (Figure 3).
Figure 3. Typical shape of ( )0
log NN
for a heterogeneous bacterial population withmin gr K<
(left) andmin gr K> (right) when a saturation bound is absent max( )N = or present max( )N < .
- Following the convention that a declining bacterial population will be completely
eradicated if there exists a time 0t> such that ( ) 1N t = , eqn. (33) implies that such
time will satisfy the equation
( ) ( )0 minln 1 0atgN K r t e + + = (37)
whose solution is
0
min
( ln )
0
min min
expln 1ProductLog
g
a N
K r
g g
aN
K r a K r
(38)
where the function ProductLog( )z is defined23
as the principal solution, w , of the
equationw
z we= .
- Estimating the dependence of minr as a function of the antibiotic concentration C(e.g.
eqn. (2)) is crucial for determining whether a bacterial population will be eventually
eradicated or grow, even if there is an initial decline due to elimination of the more
susceptible subpopulations.
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Figure 5. Data fit for 0.5 MICC= through 128 MICC= (along with 0C= ) using eqn. (33)
Table 1. Estimates of the parameters min ,r , a, for 0.5 MICC= through 128 MICC=
( s denotes standard error)
C
( MIC) minr s
(1/h)
(1/h)
a
(1/h)
0.5 0.58 0.21 0.38 0.11
2 0.89 0.04 1.3 4.1
8 0.97 0.04 11 6.4
32 1.02 0.03 12 4.9
128 1.04 0.02 12 5.4
The estimates of minr for 0.5 MICC= through 128 MICC= can be fitted by eqn. (2)
as shown inFigure 6,with corresponding parameter estimates shown inTable 2.
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Figure 6. Fit (95% confidence zone) of the kill-rate constant minr as a function of antibiotic
concentration C.
Table 2. Estimates of the parametersk
K , 50C , H for the most resistant bacterial subpopulation
Parameter Estimate standard error
kK (1/h) 1.03 0.02
50C ( MIC) 0.39 0.04
H 1.06 0.15
An important outcome of the above results is that an estimate of the antibioticconcentration Ccan be easily calculated that ensures eradication of the entire bacterial
population, including its most resistant subpopulation. Indeed, eradication is guaranteed if
min ( ) gr C K= , which, by eqn. (2), implies
1/
50 2.2
H
k g
g
K KC C
K
=
= ( MIC) (39)
as can be visualized inFigure 6. Thus, for this bacterial population, the antibiotic concentration
that would ensure complete eradication is more than double the conventional MIC, because ofthe subpopulations comprising resistant strains.
5 Discussion
A mathematical modeling framework was developed to describe the effect of antibiotics on
heterogeneous bacterial populations (entailing subpopulations of non-uniform susceptibility or
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resistance to antibiotics). The essence of the framework is captured by Theorem 2, which
generalizes and expands prior work developed for antibiotic/bacteria systems following linear
dynamics17
to the case of logistic dynamics. From a practical viewpoint, Theorem 4 offers
convenient expressions that can be easily used for parameter estimation based on
measurements of an entire bacterial population in time-kill experiments.Because of its generality,Theorem 2 makes it possible to study the outcomes of various
cases, including bimodal distributions.
Acknowledgement. The study was supported in part by the National Science Foundation (CBET-
0730454), National Institutes of Health (1R56AI111793-01) and an unrestricted grant from
AstraZeneca.
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relationships for phase-nonspecific agents.J Pharm Sci. 1971;60:892-895.
19. Wagner J. Kinetics of Pharmacologic Response I. Proposed Relationships between
Response and Drug Concentration in the Intact Animal and Man.J. Theoret. Biol.
1968;20:173-201.
20. Hill AV. The possible effects of the aggregation of the molecules of haemoglobin on its
dissociation curves.J. Physiol. 1910;40:iv-vii.
21. Nikolaou M, Schilling AN, Vo G, Chang K-t, Tam VH. Modeling of MicrobialPopulation Responses to Time-Periodic Concentrations of Antimicrobial Agents.Annalsof Biomedical Engineering. 2007;35(8):1458-1470.
22. Abramowitz M, Stegun IA, eds.Handbook of Mathematical Functions with Formulas,
Graphs, and Mathematical Tables, 9th printing. New York: Dover; 1972.
23. Corless RM, Gonnet GH, Hare DEG, Jeffrey DJ, Knuth DE. On the Lambert W Function.Advances in Computational Mathematics. 1996;5:329-359.
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Appendix A Details on experiments performed for time-kill data collection
Time-kill studies were performed with a wild-type Escherichia coli strain (MG 1655) with an
inoculum of approximately 1108CFU/ml at baseline. Apart from the placebo control ( 0)C= ,
five concentrations of moxifloxacin in Mueller-Hinton broth were used with fourfold increase
between subsequent concentrations. All concentrations were normalized to fraction/multiples
of the minimum inhibitory concentration (MIC), equal to 0.0625 g/ml , from 0.5MIC to
128MIC. The experiments were performed in a shaker water bath set at 35C. Serial samples(0.5 ml) were obtained in duplicate over 24 hours at baseline, 2, 4, 8, 12 and 24 hours; viable
bacterial burden was determined by quantitative culture. Before being cultured quantitatively,
the bacterial samples were centrifuged (10000G for 15 minutes at 4C) and reconstituted withsterile normal saline to minimize the drug carryover effect. Total bacterial populations were
quantified by spiral plating 10serial dilutions of the samples (50 ml) onto Mueller-Hinton agar
plates. The media plates were incubated at 35C for up to 24 hours in a humidified incubator,then bacterial density from each sample was enumerated visually.
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Appendix B Proof ofTheorem 1
Summation of eqn. (4) over all i yields
( )max
max
( )1 ( )
( )1 ( ) ( )
g k k
k
g
dN N t K r C N t
dt N
N tK t N t
Nm
=
=
. (40)
which is eqn. (6).
Taking derivatives of eqn. (12) with respect to time and combining it with eqns. (4) and
(6) yields
( )
( )
( ) ( )
2
max
2
max
( )1( ),
( ) ( )
( ) 11 ( )
( )
( ) ( )1 ( ) ( )
( )
( ) ( ) ( ),
k kk
g k k
kg
k k
dN N t d dNf r C t
dt dt N t N t dt
N tK r C N t
N N t
N t N tK t N t
N t N
t r C f r C t
m
m
=
=
=
(41)
Taking derivatives of the moment-generating function, eqn. (13), with respect to time,
and combining the resulting equation with eqn. (41) yields
( ) ( ) ( )
( )
( ) ( )
( )
( ),( , )( ) ( ) ( ),
( ) ( , ) ( ) ( ),
k k
k
sr C sr C k
k k
k k
sr C
k k
k
df r C t M s te e t r C f r C t
t dt
t M s t e r C f r C t
m
m
= =
=
. (42)
Taking derivatives of the cumulant-generating function, eqn. (14), with respect to time,
and combining the resulting equation with eqn. (41) yields
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( )
[ ]
( )
( , ) 1 ( , )
( , )
1( ) ( , ) ( ) ( ),
( , )
1 ( , )( )( , )
ln ( , )( )
( , )( )
ksr C
k k
k
s t M s t
t M s t t
t M s t e r C f r C t M s t
M s ttM s t s
M s tt
s
s tt
s
m
m
m
m
=
=
=
=
=
(43)
Expanding ( , )s t around 0s= using Taylor series as
1 2 3
/ / / 2 / / / 3
0 ( ) ( ) ( )
2 3
1 2 3
1 1( , ) (0, ) (0, ) (0, ) (0, ) ...
2! 3!1 1
( ) ( ) ( ) ...2! 3!
t t t
s t t t s t s t s
t s t s t s
= + + + +
= + + +
(( (( (( (44)
and using the definition of cumulants in eqn. (15) yields
2 331 2( , ) 1 1 ...2! 3!
dd ds ts s s
t dt dt dt
= + + +
(45)
and
2 3
1 2 3 4
( , ) 1 1
( ) ( ) ( ) ( ) ...2! 3!
s t
t t s t s t ss
= + + + (46)
Substituting the expansions of eqns. (45) and (46) into eqn. (43) implies
2 331 2
2 3
1 2 3 4
0
1 1...
2! 3!
1 1( ) ( ) ( ) ( ) ( ) ...
2! 3!
dd ds s s
dt dt dt
t t t s t s t s
m
=
+ + + =
((
(47)
Equating like powers of s in the above equation yields
1( )n
nd t
dt += , 1n (48)
which is eqn. (7) for 1n= , eqn. (8) for 2n= , and eqn. (9) for 3n .
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Appendix C Proof ofTheorem 2
Eqn. (48) implies
1 1
2 2
3 3
( )
/ ( )0 1 0
/ ( )0 1
0/ ( )
d dt t
d dt t
d dt t
d dt t
=
J
((((((((
. (49)
The corresponding solution of the above system of eqns. (49), which is in Jordan form, is
[ ] 1!0
( ) exp (0) (0) k k
k
k
t t t
=
= = J J , (50)
from which
2 31 11 1 2 3 42! 3!( ) (0) (0) (0) (0) ...t t t t = + + (51)
or
2 2 31 13 42! 3!
( ,0)( ) (0) (0) (0) (0) ...
s t
st t t t
sm m
=
= + + =
(52)
which is eqn. (20). Similarly,
2 31 12 2 3 4 52! 3!( ) (0) (0) (0) (0) ...t t t t = + + (53)
or
22 2 2 31 1
3 4 52! 3! 2
( ,0)( ) (0) (0) (0) (0) ...
s t
st t t t
s
=
= + + =
(54)
which is eqn. (21), and
2 31 11 2 32! 3!
( ,0)( ) (0) (0) (0) (0) ...
n
n n n n n n
s t
st t t t
s + + +
=
= + + =
, 3n (55)
which is eqn. (22).
Next, integration of the Bernoulli differential equation in eqn. (6) implies
[ ]00
0 max
( )ln ( ) ln 1 exp ( )
t
g
NN tG t K G d
N Nt t
= +
(56)
where
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0( ) ( )
t
gG t K d m = . (57)
which, by eqn. (20), becomes
( ) ( ,0)gG t K t t = + . (58)
Therefore, substituting ( )G t from the above eqn. (58) into eqn. (56) results in
0
00 max
0
0max
( )ln ( ,0) ln 1 exp ( ,0)
( ,0) ln 1 e ( ,0)g
t
g g g
t K
g g
NN tK t t K K d
N N
NK t t K M d
N
t
t t t
t t
= + + +
= + +
, (59)
which is eqn. (19).
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Appendix D Proof ofTheorem 3
Equations (27) and (26) follow immediately from eqns. (20) and (21), along with eqn. (24).
Combining eqns. (19) and (44) with eqn. (24) yields
2 212!
0
2 20 12!0
max
2 212
2 20 120
max
( )ln ( ) ( )
ln 1 exp ( ) ( )
( )
ln 1 exp ( )
g
t
g
g
t
g g
N t K t t t N
NK d
N
K t t
NK K d
N
m
m t t t
m
m t t t
= + +
+ +
= +
+ +
(60)
which immediately yields eqn. (25).
Equation (29) follows immediately from eqns. (24) and (22).
Finally, eqn. (28) follows immediately from eqn. (26) and the fact that ( ) 0tm .
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Appendix E Proof ofTheorem 4
The proof follows straightforward facts on moment- and cumulant-generating functions for
Poisson distributions and is provided for completeness.
Given eqn. (30), eqn. (13) implies
( )
( )
min
( )
0
( )
0
min
minmin
( , 0) ( ), 0
!
exp 1
exp exp 1
ktr C
k
k
kt ak r
k
at
M t e f r C
ee
k
r t e
rr t t
m
+
=
=
= +
= +
(61)
which, combined with eqn. (14), implies that the cumulant-generating function is
[ ]
( )
minmin
min
( , 0) ln ( , 0)
exp 1
1at
t M t
rr t t
r t e
m
=
= +
= +
(62)
Substituting eqns. (61) and (62) into eqn. (19) immediately yields eqn. (33).
To get a closed-form expression in terms of standard functions for the integral in the
nonlinear term of eqn. (33), consider the transformation
a
x e t= (63)
to get
( ) ( )
( )
( )
min
min min
min
min
min0
min
1
11
1
exp 1
exp ln 1
, ,
at
g
at
g g
at
g
g
ta
g
eg
r K
xe aa
e
K r r K
ze a aa
e
K rr K ate a
a a
K r e d
K r dxx x
a ax
x e dx
z e dz
e
t
t t
+
= +
=
=
=
(64)
leading to
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Bhagunde et al. Nonlinear modeling of bacterial population dynamics 25
( ) ( )
( )min
min
0
00 max
min
0
max
( )ln ( ,0) ln 1 e ( ,0)
1
ln 1 , ,
g
g
g
t K
g g
at
g
K rr Kg
ataa
NN tK t t K M d
N N
K r t e
K Ne e
a N
t
t t
= + +
= +
+
. (65)
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