I Growth of individual organisms
Assumptions:
·
Growth is expressed as the net result of material gained
through anabolic processes less material lost through catabolism
·
The gain in weight is proportional to the surface area S(t)
of the organism
·
A rate constant H is used to measure the average effect of
the anabolic processes, thus
At small time interval dt
Weight gain = HS(t) dt
·
Loss in weight is proportional to the weight of the organism
W(t)
·
A rate constant C measures the average effect of the
catabolic processes
Weight loss = CW(t) dt
The resulting growth increment dW(t) is then
dW(t) = HS(t)dt – CW(t)dt
The rate of growth in weight can be described by the
differential equation
dW/dt = HS – CW
·
The relationship between surface area and weight was assumed
based on organisms that maintain the same shape while growing
·
The surface area and the weight and volume and weight are
related to the square and cube of linear dimension
·
Example: a sphere of radius r has surface area 4pr^{2}
and the volume 4pr^{3}/3
·
The relationship in general between surface area and weight
to length is:
S = a_{2} L^{2}
W = a^{3} L^{3}
a2 and a3 are model parameters
Thus,
d(a_{3}L^{3})/dt
= H a_{2} L^{2} – Ca_{3} L^{3}
Differentiating L^{3} with respect to time
dL^{3}/dt = 3L^{2} dL/dt
By dividing both sides by 3 a_{3} L^{2}
When dL/dt = 0 that is when growth has reached the maximum
L max = Ha_{2}/Ca_{3}
Then,
dL/dt = k ( L max – L )
k = C/3
By integrating
ò dL
/ L max – L = ò k dt

log ( L max – L ) = kt + c
at t = 0 L = L
(0)

log ( L max – L ) = kt – log ( L max – L (0))
by taking exponentials of both sides
L = L max – [ L max – L (0)]
exp (kt)
Or
L(t) = L max ( 1 – A exp^{(kt)})
A = 1 – (L(0) / L max )
Since W = a_{3}L^{3}
W(t) = W max ( 1 – A exp^{(kt)})^{3}
·
A flexible empirical model for describing different patterns
of organic growth
·
Rate of growth in weight is given by the differential
equation
dW/dt
= kW/ (1m) [ (Wmax /W)^{1m} 1 ]
Wmax = maximum attainable weight
K = rate constant
Thus,
W(t) = Wmax ( 1 A* exp(kt)) ^{1/(1m)}
Flexibility is achieved
by varying the parameter m
Models for Special cases
Growth Model 
m 
Growth Equation 
Monomolecular 
0 
W(t) = Wmax (1 A* exp(kt)) 
Von Bertalanffy 
2/3 
W(t) = Wmax (1 – A * exp(kt))^{3} 
Gompertz 
1 
W(t) = Wmax exp(A * exp(kt)) 
Autocatalytic 
2 
W(t) = Wmax/(1+A* exp(kt)) 
Allometric Growth
·
Allometry is the study of the relative size of
different parts of an organism and how the parts grow in relation to each other
·
Huxley proposed a model in which the ratio of the relative
growth rates of the two components remain constant
If the two components
are named Y and X
Then,
1/Y dy/dt
= b/X dx/dt
or d log Y/dt = b * d log X/dt
b is the ratio of the relative growth rates of the
variables Y and X
Integrating both sides
log Y = log A + b * log X
Or
Y = A * X^{b}
·
The size of one component is proportional to the size of the
other component raised to some power b.
·
B is a constant with no units of measurements
·
A measures the size of Y when X is of unit size and is
therefore affected by the units of measurement