Processes 2018, 6, 134
In the case of ω = 0 the system of equations could only be reduced to the four state system of
1 −
1
f
γ r r
∗
I +
m l
ϑ l
ρ v (Wω − γ v v
∗
I )
1 − (α −L + β L )+
α −L
β L
+
m v
ϑ v
γ v v
∗
I
1 −
(α −v + β v )
β v
+
α −v
β v
+
μ r μ mr f 2 (c ∗ )
δ mr
= 0,
(50)
−γ r r
∗
I +
fm l
ϑ l
1 +
r ∗
f
1 − r ∗
f
ρ v (Wω − γ v v
∗
I )+
fm v
ϑ v
1 +
r ∗
f
1 − r ∗
f
γ v v
∗
I
+
μ r μ mr f 2 (c ∗ )
δ mr
= 0,
(51)
γ r r
∗
I +
m l
ϑ l
ρ v (Wω − γ v v
∗
I )
−r ∗
f
1 − r ∗
f
− (α −L + β L )+
α −L
β L
+
m v
ϑ v
γ v v
∗
I
−r ∗
f
1 − r ∗
f
−
(α −v + β v )
β v
+
α −v
β v
= 0,
(52)
R
chol
L σ l ρ v (Wω − γ v v
∗
I )+R
chol
v σ v γ v v
∗
I +
μ c μ h μ mh f 1 (c ∗ )
δ h δ mh
− δ c c
∗ = 0,
(53)
where
f 1 (c)=
1
1 +
κ mh
1 +
c ∗
κc
xc x h
and
f 2 (c)=
1
1 +
κ mr
1 +
c ∗
κc
xc xr .
This system was then solved numerically in Maple (Version 2016.2) informed by the parameter
values given in Table 2. Only one steady-state was determined and the values obtained were found to
concur with the steady-states obtained in Figure 3.
3.2.2. Sensitivity analysis
Sensitivity analysis was conducted to determine how the output of our model can be apportioned
to the varying sources of input. We conducted local sensitivity analysis, varying each parameter
100-fold above and below its initial value, whilst keeping all the other variables constant as shown
in Table 2. We quantitatively measured, primarily, the effect of mechanisms on the steady state
intracellular cholesterol concentrations whilst also looking for significant variations in key elements of
the system, for example extracellular LDL levels which are an indicator of risk in CVD. What follows
is a summary of our findings with more details provided in Appendix C.
Sensitivity analysis of our fully integrated model demonstrated that the respective components
(cholesterol biosynthesis and endocytosis) reproduced similar effects on intracellular cholesterol
levels as they did when considered in isolation. For instance the integrated model shows the same
competition effect between LDL and VLDL for LDLR as detailed in [13,15]; a result of VLDL having
a greater receptor binding affinity than LDL. Likewise, this behaviour can be exacerbated by the
number of receptors occluded by a bound lipoprotein and by the rate of receptor synthesis. However,
where there are limited binding sites on the cell surface, smaller LDL particles are able to bind in spaces
that larger VLDL are unable to. We found that variation in parameters affecting LDLR synthesis does
not greatly affect cholesterol levels.
We found that biosynthesis of cholesterol has a greater effect on cellular cholesterol levels than
the uptake of extracellular lipoproteins, which agrees with the known biology [3]. The concentration
of free LDLR does not greatly affect intracellular cholesterol levels, but does have a much greater
93
In the case of ω = 0 the system of equations could only be reduced to the four state system of
1 −
1
f
γ r r
∗
I +
m l
ϑ l
ρ v (Wω − γ v v
∗
I )
1 − (α −L + β L )+
α −L
β L
+
m v
ϑ v
γ v v
∗
I
1 −
(α −v + β v )
β v
+
α −v
β v
+
μ r μ mr f 2 (c ∗ )
δ mr
= 0,
(50)
−γ r r
∗
I +
fm l
ϑ l
1 +
r ∗
f
1 − r ∗
f
ρ v (Wω − γ v v
∗
I )+
fm v
ϑ v
1 +
r ∗
f
1 − r ∗
f
γ v v
∗
I
+
μ r μ mr f 2 (c ∗ )
δ mr
= 0,
(51)
γ r r
∗
I +
m l
ϑ l
ρ v (Wω − γ v v
∗
I )
−r ∗
f
1 − r ∗
f
− (α −L + β L )+
α −L
β L
+
m v
ϑ v
γ v v
∗
I
−r ∗
f
1 − r ∗
f
−
(α −v + β v )
β v
+
α −v
β v
= 0,
(52)
R
chol
L σ l ρ v (Wω − γ v v
∗
I )+R
chol
v σ v γ v v
∗
I +
μ c μ h μ mh f 1 (c ∗ )
δ h δ mh
− δ c c
∗ = 0,
(53)
where
f 1 (c)=
1
1 +
κ mh
1 +
c ∗
κc
xc x h
and
f 2 (c)=
1
1 +
κ mr
1 +
c ∗
κc
xc xr .
This system was then solved numerically in Maple (Version 2016.2) informed by the parameter
values given in Table 2. Only one steady-state was determined and the values obtained were found to
concur with the steady-states obtained in Figure 3.
3.2.2. Sensitivity analysis
Sensitivity analysis was conducted to determine how the output of our model can be apportioned
to the varying sources of input. We conducted local sensitivity analysis, varying each parameter
100-fold above and below its initial value, whilst keeping all the other variables constant as shown
in Table 2. We quantitatively measured, primarily, the effect of mechanisms on the steady state
intracellular cholesterol concentrations whilst also looking for significant variations in key elements of
the system, for example extracellular LDL levels which are an indicator of risk in CVD. What follows
is a summary of our findings with more details provided in Appendix C.
Sensitivity analysis of our fully integrated model demonstrated that the respective components
(cholesterol biosynthesis and endocytosis) reproduced similar effects on intracellular cholesterol
levels as they did when considered in isolation. For instance the integrated model shows the same
competition effect between LDL and VLDL for LDLR as detailed in [13,15]; a result of VLDL having
a greater receptor binding affinity than LDL. Likewise, this behaviour can be exacerbated by the
number of receptors occluded by a bound lipoprotein and by the rate of receptor synthesis. However,
where there are limited binding sites on the cell surface, smaller LDL particles are able to bind in spaces
that larger VLDL are unable to. We found that variation in parameters affecting LDLR synthesis does
not greatly affect cholesterol levels.
We found that biosynthesis of cholesterol has a greater effect on cellular cholesterol levels than
the uptake of extracellular lipoproteins, which agrees with the known biology [3]. The concentration
of free LDLR does not greatly affect intracellular cholesterol levels, but does have a much greater
93