Biology Reference
In-Depth Information
This provides an attractive approach to the definition of
phenotypes when an analytical solution is available.
Unfortunately, such a solution is seldom available for the
more complex non-linear systems that are characteristic of
nearly all biological systems of interest.
distinct phenotypes the system is capable of exhibiting? By
integrating information from all the steady-state solutions
and their corresponding boundary conditions, we can address
this question in the context of the system design space.
Phenotypes in System Design Space
The phenotypes corresponding to the steady-state solutions
only make sense if the solutions also satisfy the set of
inequalities required to justify the assumption of domi-
nance. This is a well-known linear programming problem
involving solution of the steady-state equations, which are
linear equations in logarithmic coordinates, along with the
corresponding set of dominance conditions, which are
linear inequalities in logarithmic coordinates
[41,42]
. The
result is a set of boundaries delimiting valid regions, which
define qualitatively-distinct phenotypes that can be visual-
ized graphically in the system design space, as shown in
Figure 15.4
.
It should be emphasized that the phenotypes in this
design space are 'generic' phenotypes for the entire class
('species') of such two-step reactions. The phenotype of an
'individual' member of this class will be located within this
space when the parameters of the individual and its envi-
ronment are specified. If an individual experiences
a change in its genotype (genetically influenced parame-
ters) or its environment (environmentally influenced vari-
ables), then the individuals location in design space will
move accordingly and, if boundaries are crossed, there will
be a qualitative change in phenotype. In any population of
individuals there will undoubtedly be some heterogeneity
in their genotypes and environments, and this will lead to
Phenotypes from the Differential Equation
Might the same approach, based on the identification of
dominant terms, be applied to the differential equation
without having to obtain its analytical solution? The answer
is yes! Examination of Eq.
(1)
suggests four qualitatively
distinct phenotypes based on the dominance among its
positive and negative terms:
dX
dt
z
½
k
S
S
k
S
X
Case 1
:
when
k
S
k
1
K
eq
G
k
S
k
p
>
and
p
>
1
dX
dt
z
½
Case 2
k
S
S
k
P
X
:
when
k
S
1
K
eq
G
k
S
k
p
<
k
P
>
and
1
dX
dt
z
½
Case 3
k
P
S
k
S
X
:
when
k
1
K
eq
G
k
S
k
P
<
S
k
p
>
and
1
dt
z
k
P
P
k
p
X
dX
Case 4
:
when
k
S
1
K
eq
G
k
S
k
P
<
k
p
<
and
1
This method of selecting one dominant positive term and
one dominant negative term in general generates a set of
nonlinear equations known as an S-system, which in steady
state reduces to a linear problem for which one can obtain
an explicit steady-state solution
[2,12]
. The steady-state
solution of the dominant differential equations is obtained
by setting the derivatives to zero and solving for the
dependent variable, and the results are seen to be exactly
the same as those obtained from the dominant terms of the
analytical solution.
This method of selecting dominant positive and negative
terms from the differential equation has several advantages
over the method of selecting dominant terms from an
analytical solution. First, the steady-state solution of the
dominant differential equations, which is a linear problem, is
much simpler than finding an analytical solution in the
general case. Second, having the differential equations based
on dominant positive and negative terms means that we also
have access to the local dynamic behavior for each of the
phenotypes. However, will this method tell us how many
FIGURE 15.4
System design space for the model in
Figure 15.3.
The
arrows show the direction of net flux in each quadrant. Case 1 (Blue): the
first reaction is operating in quasi-equilibrium. Case 2 (Green): both reac-
tions are operating in a quasi-irreversible forward direction. Case 3 (Cyan):
both reactions are operating in a quasi-irreversible reverse direction. Case 4
(Red): the second reaction is operating in quasi-equilibrium. The black dot is
the equilibrium state, log
10
ð
k
S
=
k
p
Þ
z
0
:
25, for the mutarotation of
glucose in solution
[40]
. See text for discussion.
