Modeling and parameter estimation of bacterial growth. Baranyi Model Three-Phase linear Model Richards’ Model Weibull Model Logistic Model Gompertz Model Von Bertalanffy Model

1. Modeling and parameter
estimation of bacterial
growth

2. Bacterial Growth
Four stages: Lag, Exponential,
Stationary, Death
Actual Physiological adaptation during
lag phase is too complex
Most Mathematical models focus on
log and lag phases

3. Mathematical Models
Baranyi Model
Three-Phase linear Model
Richards’ Model
Weibull Model
Logistic Model
Gompertz Model
Von Bertalanffy Model

4. Baranyi Model
Explained by Jószef Baranyi
Mathematician and statistician
Worked for the institute of food
research, UK for two decades
Further advances in food microbiology
would be impossible without a
dedicated mathematician.
Focuses on lag phase calculations

5. Modeling Population Growth in
Baranyi Model
A non- autonomous model where an
α(t) factor, a so-called adjustment
function describes the transition from
the lag to the exponential phase.
dx/dt = α(t)μ(x) (0≤t<∞; 0<x)
x(0) = x0 where 0<x0<xmax
0 ≤α(t)≤1 (0≤t<∞) α(t) → 1 monotone
increasingly as t → ∞
α(t) is a sufficiently smooth function. It
can depend on the pre-inoculation
environment

6. Functions for α(t):
α(t) can be expressed using many
functions to suit different conditions.
Using Hill function to model Adaptation
period:
α(t)= tn/(λn+tn)
where n is +ve & λ is a parameter which is
a suitable definition of lag time.
Further, this function has an inflection
point: Λn= λn√(n-1)/(n+1)

7. Functions for α(t):
(cont.)
Using Michaelis-Menten kinetics:
α(t)= P(t)/Kp+P(t)
KP is the Michaelis-Menten constant and
after inoculation, the accumulation of P(t)
follows a first order kinetic process.
dP/dt = vb P where vb is the characteristic of
the environment.

8. Estimating the distribution
parameters in Baranyi Model:
To quantify the physiological state of the
initial population α is introduced.
 α= e -μ λ
The physiological state of the inoculum,
however, is equal to the arithmetical
mean of the physiological states of the
individual cells, the αi=e -μτi quantities.
This is physiological state theorem
ANOVA is developed using this theorem.
 Initial number of cells (N0) in a well
follows a Poisson Distribution.

9. Three-phase linear
model

10. THREE PHASE LINEAR
Imposes a horizontal line fits on the lag and
stationary periods and uses least square
regression to allocate observations to three
phases as well as to the co-ordinates of the
three lines themselves.
This extension of early graphical methods of
microbiologists is mechanistic and very
simple and is a fully adequate for use as a
primary model for support multi-parameter
environmental modeling of bacterial growth.

11. MODELING BACTERIAL GROWTH IN THREE-PHASE LINEAR
MODEL:

12. MODELING BACTERIAL GROWTH IN THREE-PHASE LINEAR
MODEL: (CONT.)
Most food micro- biology applications are
not overly interested in the stationary
phase. In reality, if the stationary phase is
reached, the food is either spoiled if the
microorganism is non-pathogenic or a
threat to public health if it is a pathogenic
species.
Liquid cultures, particularly when they are
agitated, have rather rapid transition
between exponential and stationary
growth.

13. Richards’
Model

14. Richards’ Model:  The generalised logistic curve or function, also
known as Richards' curve is a widely used and
flexible sigmoid function for growth modelling,
extending the well-known logistic function.
 It has six parameters:
A: the lower asymptote;
K: the upper asymptote. If A=0 then K is called the
carrying capacity;
B: the growth rate;
ν>0 : affects near which asymptote maximum growth
occurs.
Q: depends on the value Y(0)
M: the time of maximum growth if Q=ν

15. A particular case of Richard's function is:
which is the solution of the so-called Richard's
differential equation (RDE):
with initial condition
Where
provided that ν > 0 and α > 0.

16. Parameter Estimation:
When estimating parameters from
data, it is often necessary to compute
the partial derivatives of the
parameters at a given data point t

17. Weibull
Model:

18. The Weibull growth model is described
by the equation:
where l = length, (or weight, height, size)
and t = time.
The four parameters are:
β, is the lower asymptote;
L∞, is the upper asymptote;
k, is the growth rate and
δ, is a parameter that controls the x-ordinate
for the point of inflection.

19. The point of inflection on the x axis lies
at:
If δ = 1 the Weibull is a
simple exponential
growth curve.

20. ω(t) = (β0-β1exp(-β2tβ3)) + ε
Parameters can be estimated by
partial
derivatives of ω with respect to β0, β1,
β2,
β3
β0 is the asymptote or the potential
maximum of the response variable;
β1 is the biological constant;
β2 is the parameter governing the rate
at which the response variable
approaches its potential maximum;

21. Logistic
Model:

22. Logistic Model:
The logistic growth curve (sometimes called the
Verhulst model as it was first proposed as a
model of population growth Pierre Verhulst
1845, 1847) is one of the simplest of the S-shaped
growth curves.
A biological population with plenty of food,
space to grow, and no threat from predators,
tends to grow at a rate that is proportional to
the population -- that is, in each unit of time, a
certain percentage of the individuals produce
new individuals. If reproduction takes place
more or less continuously, then this growth
rate is represented by this distribution.

23. Modeling Growth:
where t is time, l is length (size), K is the
growth rate and delta a term which
expresses the rate at which growth
declines with size.
After integration and some rearrangement
we arrive at the 3 parameter logistic
growth curve: I is the age at the inflection
point and L∞ is the upper asymptote
(maximum size reach after infinite
growing time).

24. The 3 parameter logistic has a lower asymptote of 0. The
point of inflection on the y-axis occurs at
This last formula states that the point of inflection is
always at at 50 % of the asymptotic size (L∞). This does
not hold true for all growth processes. You should
consider using the Logistic growth curve to model
sigmoid growth processes in which the point of
inflection is approximately 1/2 of the maximum possible
size.
If a non-zero asymptote is required then the 4 parameter
version of the equation is required this is expressed by
the equation:
where a is the lower asymptote and d is a shape
parameter that determines the steepness of rising
curve.

26. Gompertz
Model:

27. Modeling Growth:
The Gompertz curve was originally
derived to estimate human mortality
by Benjamin Gompertz (Gompertz, B.
"On the Nature of the Function
Expressive of the Law of Human
Mortality, and on a New Mode of
Determining the Value of Life
Contingencies.") Charles
Winsor (1932) presented an early
description of the use of this equation
to describe growth processes.

28. where K is the growth rate and L∞, termed 'L infinity', is the
asymptotic length at which growth is zero..
Integrating this becomes:
where t is age and I is the age at the inflection point.
The equation above is the 3 parameter version of the
Gompertz growth curve (see below for an example
plot). Growth II can also fit the 4 parameter version:
in which A is the lower asymptote (see below for an
example plot) and B is the upper asymptote minus A.

29. The point of inflection on the y-axis
occurs
at
This last formula states that the point of
inflection is always at about 36.8 % of
the asymptotic size (L∞). This does not
hold true for all growth processes. You
should consider using the Gompertz
growth curve to model sigmoid growth
processes in which the point of
inflection is approximately 1/3 of the
maximum possible size.

30. von
Bertalanffy
Model

31. von Bertalanffy derived this equation in 1938
from simple physiological arguments. It is
the most widely used growth curve and is
especially important in fisheries studies.
In its simplest version the so-called von
Bertalanffy growth equation is expressed
as a differential equation of length (L) over
time (t):
K is the growth rate and
L∞, termed 'L infinity' in fisheries science, is
the asymptotic length at which growth is
zero.

32. Integrating this becomes:
The parameter t0 is included to adjust the
equation for the initial size of the
organism and is defined as age at
which the organisms would have had
zero size. Thus to fit this equation you
need to fit 3 parameters (L∞, K and t0 )
by nonlinear regression.

33. To fit this curve we must therefore
estimate 3 parameters, L∞, K and t0.
While this was once done graphically, it
is now accomplished using the
Levenberg-Marquardt Method for non-linear
regression.

34. And Finally!!!