Processes 2018, 6, 134
internalised and esterified, leading to an abundance of free receptors on the cell surface and LDLR and
cholesterol being produced via their respective biosynthetic pathways
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
m ∗
h
m ∗
r
h ∗
l ∗
E
l ∗
RB
l ∗
I
v ∗
E
v ∗
RB
v ∗
I
r ∗
f
r ∗
I
c ∗
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
μ mh
δ mh
⎛
⎜
⎜
⎝ 1+
⎛
⎜
⎜
⎝ κ mh
⎛
⎜
⎜
⎝ 1+
⎛
⎜
⎝
μc μ h m ∗
h
δ h δc
κc
⎞
⎟
⎠
4 ⎞
⎟
⎟
⎠
⎞
⎟
⎟
⎠
3 ⎞
⎟
⎟
⎠
μmr
δmr
1+κmr(1+ c ∗
κc )
4
μ h m ∗
h
δ h
0
0
0
0
0
0
P
γrr ∗
I
β 0
μrm ∗
r
γr(1− f )
μcμ h m ∗
h
δ h δc
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
,
(49)
where the ∗ notation indicates steady-state. This result was also verified numerically.
Figure 3. Numerical simulation of Equations (36)–(48). Initially mRNA levels increase in response
to zero cholesterol in the system, which leads to an increase in HMGCR and internal receptor levels.
VLDL and LDL bind to receptors and are internalised where cholesterol is extracted. The increase in
HMGCR and extraction of cholesterol from internalised lipoproteins cause intracellular cholesterol
concentrations to increase. Damped oscillations can be seen as HMGCR mRNA responds to changing
cholesterol concentrations and the receptor mechanism responds accordingly. Following this initial
transient behaviour, the molecular and cholesterol parts of the system settle to a relatively stable
steady-state whilst VLDL and LDL continues to be extracted from the extracellular environment until a
final steady-state is reached.
92