Growth Factors, 2002 Vol. 20 (4), pp. 155–175
A Mathematical Model for the Role of Cell Signal Transduction
in the Initiation and Inhibition of Angiogenesis
HOWARD A. LEVINEa,*,†, ANNA L. TUCKERa,†,‡ and MARIT NILSEN-HAMILTONb,{
a
Department of Mathematics, Iowa State University, Ames, IA 50011, USA; bDepartment of Biochemistry, Biophysics and Molecular Biology, Iowa State
University, Ames, IA, USA
(Received 19 July 2002; Revised 21 November 2002)
Neovascular formation can be divided into three main stages (which may be overlapping): (1) changes
within the existing vessel, (2) formation of a new channel, (3) maturation of the new vessel.
In two previous papers, [Levine, H.A. and Sleeman, B.D. (1997) “A system of reaction diffusion
equations arising in the theory of reinforced random walks” SIAM J. Appl. Math. 683– 730;
Levine, H.A., Sleeman, B.D. and Nilsen-Hamilton, M. (2001b) “Mathematical modelling of the onset
of capillary formation initiating angiogenesis.” J. Math. Biol. 195– 238] the authors introduced a new
approach to angiogenesis, based on the theory of reinforced random walks, coupled with a Michaelis –
Menten type mechanism which views the endothelial vascular endothelial cell growth factor (VEGF)
receptors as the catalyst for transforming into a proteolytic enzyme in order to model the first stage.
It is the purpose of this paper to present a more descriptive yet not overly complicated mathematical
model of the biochemical events that are initiated when VEGF interacts with endothelial cells and
which result in the cell synthesis of proteolytic enzyme. We also delineate via chemical kinetics, three
mechanisms by which one may inhibit angiogenesis (inhibition of growth factor, growth factor receptor
and protease function).
Keywords: Neovascular formation; Angiogenesis; Promoter mechanisms; Inhibitor mechanisms
INTRODUCTION
The cell, as the smallest denominator of living organisms
capable of independent life, takes in and metabolizes
nutrients that are used for its maintenance, movement and
reproduction. In the context of a multicellular organism,
cells must also use nutrients to synthesize signals that are
released and that regulate the activities of other cells.
An extraordinary number of metabolic pathways, signal
transduction cascades and other regulatory elements
are required for a cell to synthesize the macromolecules
necessary for life and to coordinate its own activities. One
has only to confront a diagram as can be found on page
415 of Voet and Voet (1995) to obtain a sense of the
complexity of cellular metabolism.
Many more synthetic and regulatory mechanisms
are involved in maintaining the tightly coordinated and
cooperating multicellular entity that we refer as the body.
Deviation in this cellular community called the body from
the normal tightly organized and cooperating mode can
result in one of the many forms of degeneration known as
disease.
In tumor angiogenesis also, there are several pathways
by which an avascular tumor may induce the endothelial
cell lining of a nearby capillary to break through the
capillary lamina and grow toward the tumor. See Folkman
(1992), and the review article of Z.C. and Y. (1999) for
example. Two excellent review articles by Paweletz and
Knierim (1989) and Rakusan (1995) provide the reader
with an introduction to tumor angiogenesis.
*Corresponding author. Department of Mathematics, 410 Carver Hall, Ames, IA 50011, USA. Tel.: þ1-515-294-8145. Fax: þ 1-515-294-5454.
E-mail: halevine@iastate.edu
†
The authors were supported by NSF grant DMS-98-03992.
‡
E-mail: atucker@iastate.edu
{
E-mail: marit@iastate.edu
All figures in this article were generated using Matlab 6.0. In Figures 2AP1, 4AP1, 8AP1, the precision on the z-axis is greater than the resolution
allowed by the Matlab graphics package. The package allows a display of no more than four or five significant digits before rounding or rescaling. The
result is to be expected since S0 represents the total concentration of available resources and far exceeds what is necessary to open the capillary wall.
ISSN 0897-7194 print q 2002 Taylor & Francis Ltd
DOI: 10.1080/0897719031000084355
156
H.A. LEVINE et al.
It is the purpose of this article to study one of the pathways
by which growth factor is “converted” into protease, namely
the MAP kinase pathway, and to present a somewhat more
detailed kinetic mechanism for the “conversion” of growth
factor into protease by endothelial cells than the simplified
mechanism that was utilized by Levine and Sleeman (1997)
and Levine et al. (2001b; 2003). The mechanism on which
we focus is illustrated in Fig. 2 of Eckhard (1999) and
which may be viewed at www.hosppract.com/issues/1999/
01/eckhardt.htm. A molecule of vascular endothelial cell
growth factor (VEGF) binds to a cell surface receptor and
initiates a cascade of events within the cell cytoplasm and the
cell nucleus which results in the production of several
molecules of proteolytic enzyme as well as in a receptor of
the same type that was bound to the growth factor in
the first place. Two excellent articles by Kyriakis and
Avruch (2001) and Takai et al. (2001) review the MAPkinase pathway.
The MAP kinases constitute an important signaling cascade that regulates transcriptional and metabolic activity.
This group of protein kinases is organized as a multi-protein
complex associated with one or more scaffold proteins.
Activation of the first kinase, Raf, is achieved by the
monomeric G-protein Ras, which is in turn activated by
growth factor receptors that are bound to their respective
growth factor ligands. Activated Raf phosphorylates and
activates ERK (MAP kinase–kinase) which then phosphorylates and activates MAP kinase. MAP kinase phosphorylates and activates transcription factors such as AP-1 that
interacts with DNA and stimulates specific gene expression.
The full mechanism (or at least a substantial part of it
that is already understood) is outlined in the appendix to
this paper. The reader will see that the kinetics discussed
in the appendix lead to a system of at least thirty ordinary
differential equations that describe the chemical kinetics
of this pathway. Several difficulties arise with the use of
such a system of ordinary differential equations to replace
the simple model used by Levine and Sleeman (1997) and
Levine et al. (2001b; 2003). First, the size of the system
itself is formidable. Secondly, and more importantly, the
kinetic constants involved at most steps are unknown.
There are a number of papers that study the kinetics of
various pieces of this pathway and how they influence cell
signaling. For example, Levchenko et al. (2000) focused
their attention on how the scaffold proteins affect protein
1,2
kinase signaling.
THE PROPOSED MECHANISM
Levine and Sleeman (1997) and Levine et al. (2001b; 2003),
presented a simplified model for the interaction of angiogenic
growth factors such as VEGF with growth factor receptors
on the surface of endothelial cells as follows (Fig. 1).
If V denotes a molecule of angiogenic factor (substrate)
and R denotes some receptor on the endothelial cell wall,
they combine to produce an intermediate complex, RV
which is an activated state of the receptor that results in the
production and secretion of proteolytic enzyme, C, and a
modified intermediate receptor R0 . The receptor R0 is
subsequently removed from the cell surface after which it
is either recycled to form R or a new R is then synthesized
by the cell. It then moves to the cell surface.3
Likewise, the proteolytic enzyme molecule, C, moves
to the exterior of the cell surface where it degrades the
laminar basement membrane leaving products F0 by acting
as a catalyst for fibronectin degradation. The products F0
need not concern us here.4 We used classical Michaelis–
Menten kinetics for this standard catalytic reaction.
1
The model presented here views the endothelial cell as the minimal subdivision of a tissue that participates in the process of building a capillary.
As such, the cell is the recipient of a growth factor-generated signal to which it responds by movement. Movement is driven by the release of active
protease. The model takes advantage of the fact that any biochemical pathway, no matter how complex, which is defined only by the input (e.g. growth
factor–receptor complex) and the product (e.g. active protease), can be described by a single rate constant that is defined by the rate-limiting step in the
pathway. The model is not designed to describe or to contribute to an understanding of signal transduction. Although currently descriptive, is expected to
contribute to an understanding of cell behavior. It is also anticipated by the authors that the model might become more complex with the addition of more
elements such as other growth factors and inhibitors of angiogenesis as these become better defined by experimental analysis. However, the strength of
the model is that, in its simplicity and without the inclusion of many “correction factors”, it describes many aspects of angiogenesis quite faithfully.
2
The model as presented here, does not yet include the possibility of stem cells adding to the source of new endothelial cells within the tumor. We are
currently developing this aspect of the model. Like experimental research, model building is incremental. The current model provides a flexible platform
for the incorporation of currently known and future discoveries of cellular events in angiogenesis with the hope and expectation that, as these are
incorporated, the model will more accurately describe the biological reality and will provide important predictions that can help the development
of angiogenesis research.
3
Whereas, the model would seem to suggest that a single form of growth factor interacts with a single receptor type, that is not the intent. There are
many isoforms of VEGF, other growth factors, and many receptors that can signal endothelial cells to undergo angiogenesis. Although it could be
expanded to include specific growth factor–receptor interactions, we have instead decided to maintain a simpler mathematical form in which each
element of the model is viewed as a weighted composite of all constituents of that type. For example, all growth factors that induce angiogenesis and their
cognate receptors are represented by the terms V and R, respectively. Similarly, fibronectin (F) represents a composite of extracellular matrix molecules
such as laminin and collagen for example. The equilibrium constants k1 and k21 can be viewed as representing a composite of growth factor– receptor
interactions. In future versions of the model when more is known about local concentrations of particular growth factors and their receptors in specific
tissues, the model might be expanded, for particular situations such as in different tissues, to include the most predominant growth factors.
Although events such as receptor dimerization or oligomerization and growth factors that exist as disulphide linked homodimers are not included
in the model, knowledge of these events influences the effective concentration of receptor and growth factor that is entered into the calculations.
When more complete data is available regarding receptor and growth factor concentrations in tissues in which angiogenesis occurs, we expect to
update our input values based on the current mechanistic understanding of the particular growth factors, their receptors and how they interact.
4
Some of these products contribute to a negative feedback loop that should ordinarily be included in the kinetic equations we derive here.
However, since we understand that the concentrations of these products contribute to the inhibition of angiogenesis, including terms involving
them in our kinetics adds nothing to our understanding of the mathematical processes involved and only further complicates the dynamical
equations. We are planning a paper on plasmin– plasminogen activator/PAI dynamics in which will address this particular point.
A MATHEMATICAL MODEL
157
FIGURE 1 Schematic diagram for the mechanisms (2.5)– (2.8). In order to simplify the mechanisms, the transport protein Qt was not included in the
mechanism so that C0 and C are the same species in the kinetics for this system. The species notation is given in Table I.
The point of view there was that the receptors at the
surface of the cell function the same way an enzyme
functions in classical enzymatic catalysis. In symbols,
k1
k2 0
k2
where n is the number of protease molecules produced in
response to a single molecule of growth factor. (This
would lead to the production –consumption equations for
protease and growth factor of the form:
V þ R Y RV; RV ! C þ R0 ; R0 ! R;
kð21Þ
ð2:1Þ
l1
l2
C þ F ! CF; CF ! F 0 þ C:
The mechanism was simplified by combining steps
(2) and (3) in the above mechanism as follows:
k1
k2
V þ R Y RV; RV ! C þ R;
kð21Þ
l1
ð2:2Þ
l2
0
C þ F ! CF; CF ! F þ C:
The principle problem with this mechanism is that it
does not reflect the fact that a single molecule of growth
factor signals a cascade of intracellular events that result in
a cellular response that results in several (perhaps
hundreds) of molecules of protease.5 One might be
tempted to replace Eq. (2.2) by
k1
k2
V þ R Y RV; RV ! nC þ R;
kð21Þ
l1
ð2:3Þ
l2
C þ F ! CF; CF ! F 0 þ C:
d½C
d½V
¼ nk2 ½RV ¼ 2n
:
dt
dt
ð2:4Þ
where [Z ] denotes the concentration of species Z in
micro-moles per liter (micromolarity)).
However, such a mechanism is not stoichiometric.
On the other hand, the more detailed mechanism
outlined in the appendix suffers from the drawbacks
mentioned above.
In order to have a relatively simple mechanism that
is stoichiometric, we need to introduce into the model a
source of supply of amino acids from which the
ribosome can direct the assembly into protease and into
cell receptors using the respective mRNA’s as
templates. These amino acids come from the blood
plasma that bathes the endothelial cells on their
lumenal side or through the basement membrane and
from the surrounding tissue on their ablumenal side via
the cell surface transport proteins or directly from those
already found in the cytoplasm.
To describe the mechanism symbolically, we use
the notation in Table I. With these definitions in mind
5
An additional complicating factor is that this mechanism, like most kinetic mechanisms, does not take into account the local environment. Here the
complicating issue is the fact that the number n may depend upon the concentration of growth factor at the cell surface. For example, it was observed that
the concentration of protease is a bimodal function of the concentration of growth factor say ½C ¼ fð½VÞ (Unemori et al., 1992). For the simulations we
present here, we shall assume very low concentrations of growth factor. Then the number n ¼ f0 ð0Þ and can be numerically interpolated from the data
(for example Unemori et al. (1992)). However, this approximation cannot be employed near the tip of a growing capillary as it marches toward a source
of high concentration of growth factor since f([V ]) is not a linear function of [V ] for large [V ].
158
H.A. LEVINE et al.
TABLE I Notation
Species
Notation
Amino acids
Transport protein
Amino acid—transport protein complex
Amino acids
Vascular endothelial growth factor
Receptor–VEGF complex
Invaginated and degraded VEGF
G-protein activated [RV ] complex
Surface receptor
Total available extracellular protease
Basement lamina protein
X
Pt
X Pt
Y
V
RV
V*
R^
R
C
F
k1
k2
kð21Þ
kð22Þ
ð2:5Þ
k3
½X ¼ sðtÞ
½Pt ¼ pðtÞ
½XPt ¼ lðtÞ
½Y ¼ yðtÞ
½V ¼ vðtÞ
½RV ¼ mðtÞ
½V* ¼ v* ðtÞ
^ ¼ r^ ðtÞ
½R
½R ¼ rðtÞ
½C ¼ cðtÞ
½F ¼ f ðtÞ
Tissue and plasma
Trans-membrane
Cell cytoplasm
Cell cytoplasm
Extracellular matrix
Cell surface and lipid bilayer
Cell cytoplasm
Cytoplasm
Trans-membrane
Extra cellular matrix
Basement lamina
k5 ðtÞ ¼ kHðt 2 t0 Þ
which describes the transport of amino acids to and from
the exterior of the cell through the lipid bilayer and into
the cell cytoplasm6 (The technical term for this is
“facilitated diffusion”). Next, growth factor interacts with
the EC cell receptor
k4
^
V þ R Y RV; RV ! V* þ R;
Source/location
It is important to note that all of the k0 s in the above
mechanism are constant except for k5. We take k5 to be
time dependent and of the following form:
we write:
X þ Pt Y XPt ; XPt Y Y þ Pt ;
Concentration
ð2:9Þ
where H(x) denotes the Heaviside function
(
1; if x $ 0;
HðxÞ ¼
0; if x , 0
and where t0 is the mean time of the endothelial cell
protease response to growth factor. It can be estimated
that the EC begin the production of protease
approximately 15 –20 h after the growth factor ligand
binds with the cell receptor. In the Appendix below we
have attempted to list some of the mechanistic steps
responsible for this delay.8
Finally this protease degrades the proteins of the
basement lamina and the extracellular matrix:
ð2:7Þ
kð23Þ
which roughly describes the endocytosis of the RV complex
with the resultant cellular destruction of growth factor and
^
the production of a G-protein activated intermediate, R:
^
Next, R initiates events that result in the assembly of
intracellular amino acids into intracellular protease which
is then converted to extracellular protease.7
0
k5 ðtÞ
k5 ðtÞ
R^ þ Y ! nC0 þ R; C0 ! C:
l1
l2
C þ F Y CF; CF ! F 0 þ C:
ð2:10Þ
lð21Þ
We combine these two steps in a single equation:
k5 ðtÞ
R^ þ Y ! nC þ R:
where the total available protease is given by
ð2:8Þ
½Ctotal ¼ ½C þ ½CF ¼ ½C0 :
6
The mechanism (2.5) can be viewed as a shortened version of a mechanism for the two way flux of cationic amino acids across a plasma membrane
discussed by White and Christensen (1982; 1983). In the first of these references the authors report that, on the basis of kinetic analysis, “the inward and
outward transport of cationic amino acids through the plasma membrane of fibroblasts and HTC cells is mediated mostly by a single saturable transport
system. . ..” They also report that “the mediated arginine influx is half maximally saturated at an external substrate concentration of 0.1– 0.2 as high as
the apparent intracellular concentration that half maximally saturates the efflux.” This leads us to an estimate for K e ¼ k1 k2 =kð21Þ kð22Þ < 15:0 for the
value of the equilibrium constant when Eq. (2.5) is in equilibrium. The mechanism used by White and Christensen (1982; 1983) which describes the
iso-uni-uni transport system across the cell membrane has the form:
k1
kð23Þ
k2
kð21Þ
k3
kð22Þ
X þ P1t Y Z; P1t Y P2t ; Z Y Y þ P2t :
ð2:6Þ
(See, Segel (1975) for a very thorough treatment of the whole issue of enzyme kinetics in addition to the present mechanism). Here Z ¼ XP1t ¼ YP2t
represents the intermediate while the second equation represents the exchange of extracellularly oriented transport protein P1t with intracellularly
oriented protein P2t (Notice that in the first and third of these reactions, the arrows pointing from the extra-cellular to the intracellular side of the
membrane have rate constants with positive subscripts, while those pointing in the other direction have negative subscripts. In the second, the convention
is reversed). When Eq. (2.6) is in equilibrium, the equilibrium constant is K e ¼ k1 k2 k3 =kð21Þ kð22Þ kð23Þ ¼ ½Y=½X: Mechanism (2.5) is is a condensed
version of Eq. (2.6). When the forward and reverse rate constants in the second step of Eq. (2.6) are the same, this expression for Ke reduces to ours.
7
The model includes the requirement for activation of proteases from latent forms. This is because, as discussed previously, a biochemical
pathway can be represented by the single rate-limiting step (k2) that occurs between the input (growth factor–receptor complex) and output
(active protease). The rate-limiting step could be the very last in this pathway to generation of active protease which is activation of latent
protease, or it could be an earlier step such as rate of protein synthesis or rate of secretion. The model does not limit the choice of which step is
rate limiting. This will be determined by the results from experimental analyses.
8
The use of generic intermediates such as Y and the introduction of “delays” such as we have done above is not uncommon in modeling
biochemical kinetics. In particular, Lev Bar-Or et al. (2000), have presented a kinetic model for the p53-Mdm2 feedback loop which employs
both of these devices.
A MATHEMATICAL MODEL
If the total protease were constant, then we could write
(in the absence other sources of basement lamina proteases)
d½F
K cat ½C0 ½F
¼2
dt
K m þ ½F
where Michaelis– Menten kinetics applies. However, in
our case, [C ]0 is not fixed. We will replace it by the total
concentration of protease that results from step (2.10) as a
function of time. Levine et al. (2001a) have carried out a
careful justification for this. (K cat ¼ l2 and K m ¼
ðlð21Þ þ l2 Þ=l1 Þ:
The variables in Table I that are of interest to us are
v; c; r; r^; f : In addition we will need notation for the
endothelial cell density. This we introduce later.
A set of chemical equations such as (2.5) –(2.8)
generally does not take into account other influences on
species concentrations, such as crowding or dispersion.
For example, the transport proteins and other components
of an endothelial cell will become more concentrated in
three-dimensional space as the local cell density increases.
To account for this geometric effect, we will include a
term C r 0 ðtÞ in the rate equation for the transport protein
density. This is the number of additional micromoles per
liter per unit time by which the concentration of transport
protein is increased due to crowding or dispersion (Levine
et al., 2001a). We assume that Cr ð0Þ ¼ 0:
The Law of Mass Action applied to the chemical
equations (2.5) –(2.8) yields9
›s
¼2k1 sðtÞpðtÞþkð21Þ lðtÞ;
›t
›p
¼2kð22Þ yðtÞpðtÞ2k1 sðtÞpðtÞþðkð21Þ þk2 ÞlðtÞþC r 0 ðtÞ;
›t
›l
¼k1 sðtÞpðtÞþkð22Þ yðtÞpðtÞ2ðkð21Þ þk2 ÞlðtÞ;
›t
›m
¼k3 rðtÞvðtÞ2ðkð23Þ þk4 ÞmðtÞ;
›t
›y
ð2:11Þ
¼k2 lðtÞ2k5 ðtÞ^r ðtÞyðtÞ2kð22Þ yðtÞpðtÞ;
›t
›v
¼kð23Þ mðtÞ2k3 rðtÞvðtÞ;
›t
›r
¼k5 ðtÞ^r ðtÞyðtÞþkð23Þ mðtÞ2k3 rðtÞvðtÞ;
›t
›r^
¼k4 mðtÞ2k5 ðtÞ^r ðtÞyðtÞ;
›t
›c
¼nk5 ðtÞ^r ðtÞyðtÞ2 mcðtÞ
›t
159
(The quantities of amino acids resulting from growth
factor degradation are assumed to be negligible in
comparison to the quantities of amino acids from the
surrounding tissue and blood plasma needed to assemble
protease. Thus the rate equation for this quantity is omitted
from the above list).
In the last of the equations above we have included
the term 2 mc which models protease decay. The half
life, ln 2/m, is fairly small.
Suppose the initial value of growth factor is v0 ¼ 0:
Then the above mechanism will not induce any protease.
In this case, we may assume that the mechanism of
amino acid transport, Eq. (2.5) is in equilibrium and the
concentrations of X; Y; P; L are all constants. Calling them
s0, y0, p0, l0 we have
y0 ¼
k 1 k 2 s0
¼ K e s0 ;
kð21Þ kð22Þ
ð2:12Þ
ðk1 s0 þ kð22Þ y0 Þp0 k1 s0 p0
¼
:
l0 ¼
kð21Þ þ k2
kð21Þ
Representative values of the rate constants and
ptotal ¼ p0 þ l0 ; s0 may be found from the literature.
Some of them are given in our simulations below.
In all cases, when cð0Þ ¼ 0 we have the following
conservation laws where r0 is the number of available
growth factor receptors:
ðt
1
0
0
sðtÞ þ yðtÞ ¼ s0 þ y0 2 cðtÞ þ m cðt Þdt ;
n
0
pðtÞ þ lðtÞ ¼ p0 þ l0 þ C r ðtÞ;
ð2:13Þ
r^ðtÞ þ rðtÞ þ mðtÞ ¼ r^0 þ r 0 þ m0 ;
r^ðtÞ þ vðtÞ þ mðtÞ ¼ r^0 þ v0 þ m0 2
ðt
1
cðtÞ þ m cðt0 Þdt0
n
0
where m0 ; r^0 are constants of integration.10
Our analysis will be simplified considerably if we
assume that the concentrations of the intermediates X Pt
and RV are nearly constant, i.e. that X is in excess and
either R is excess relative to V or vice versa, so that
lðtÞ ¼
sðtÞpðtÞ yðtÞpðtÞ
rðtÞvðtÞ
þ ð22Þ ; mðtÞ ¼
;
1
Km
Km
K 2m
ð2:14Þ
9
Partial time derivatives are used here in anticipation of the material in the sequel in which the concentrations will also exhibit dependence on spatial
variables.
10
In the case that n depends upon the local concentration of of v, n ¼ nðvÞ say, then the quantity
ðt
1
cðtÞ þ m cðt0 Þ dt0
n
0
must be replaced by the expression
ðt cðtÞ
mcðt0 Þ n0 ðvðt0 ÞÞvt ðt0 Þcðt0 Þ0
þ
þ
dt0
0
2
0
nðvðtÞÞ
n ðvðt ÞÞ
0 nðvðt ÞÞ
in Eq. (2.13) and every equation containing the former expression which follows from this. That n depends on v can be seen experimentally in the articles
of Unemori et al. (1992) and Wang and Keiser (1998). From the point of view of an individual cell, one cannot assert that n(v) molecules of protease will
be produced for a single molecule of growth factor. However, the reader should understand, that we are dealing with ensemble averages here, as is the
case in all cases involving chemical kinetics. That is, we are not dealing with individual cells but with cell densities and viewing cells in the same spirit as
one views electrons, i.e. as a probability density.
160
H.A. LEVINE et al.
ð22Þ
where we have set K 1m ¼ ðkð21Þ þ k2 Þ=k1 ; K m
¼
2
ðkð21Þ þ k2 Þ=kð22Þ ; and K m ¼ ðk4 þ kð23Þ Þ=k3 : (We are
assuming reaction (2.5) involving transport proteins is of
Michaelis –Menten type in that the concentration of the
intermediate does not vary much in time. We also assume
that Eq. (2.7) enjoys a similar property in so far as
the first equation in that pair is concerned. This involves
the assumption that the growth factor is present in very
low concentrations relative to the number of available
receptors). (Notice that Eq. (2.14) forces the choice
m0 ¼
r 0 v0
K 2m
ð2:15Þ
upon us).
Some routine algebra leads to the following system
of algebraic and differential equations:
ðt
1
cðtÞ þ m cðt0 Þ dt0 ;
y 2 y0 ¼ ðs0 2 sÞ 2
n
0
zðtÞ ¼
sðtÞ
yðtÞ
þ ð22Þ ;
1
Km Km
r ¼ r 0 þ v 2 v0 þ
ðt
1
cðtÞ þ m cðt0 Þ dt0 ;
n
0
However, we want to model the impact of these
variables on cell movement and tissue degradation, i.e. on
h, f where h is the cell density and f is the density of the
capillary wall proteins.
Thus two problems remain. First, we must relate the
crowding of the transport proteins to the endothelial cell
density and second, we need two more equations that
relate h, f to s; y; c; v; r; r^:
The first task is relatively easy. We write p0total ¼
p0 þ l0 ¼ d0 h0 where p0total is the total available number
of transport proteins per cell. d0 is the density (in micromoles per liter) of transport proteins on the surface of
endothelial cells in a normal capillary. We then write
p0 þ l0 þ C r ðx; tÞ ¼ dðx; tÞhðx; tÞ < d0 hðx; tÞ:
The second task is more complex. We need the rate
equation for the protein (loosely designated as fibronectin)
density of the basement lamina. This is a complex
structure made up primarily of fibronectin and various
other collagens. The differential equation used by Levine
and Sleeman (1997) and Levine et al. (2001b) that
describes the time evolution of this protein density is:
›f
4f
¼
›t T f
r 0 þ v þ K 2m
ðv0 2 vÞ
ð2:16Þ
K 2m
ðt
K 2m þ v 1
0
0
cðtÞ þ m cðt Þ dt ;
2
n
K 2m
0
›s
k2 sðtÞ kð21Þ yðtÞ p0 þ l0 þ C r ðtÞ
¼ 2 1 þ
;
ð22Þ
1 þ zðtÞ
›t
Km
Km
r^ ¼
12
f
fM
h
K cat cf
2
h0 K m þ f
ð2:18Þ
where now K cat ¼ l2 and K m ¼ ðlð21Þ þ l2 Þ=l1 are the
kinetic constants arising from the second pair of kinetic
equations in (2.3). fM is the density of the BL and Tf is the
logistic growth time.11
The second missing equation is a differential
equation that describes the time evolution of h. This is a
somewhat involved story.
›v
k4
›c
¼ 2 2 vr; ¼ nk5 ðtÞ^ry 2 mc:
›t
›t
Km
where we have suppressed the time variable. We also take
r^0 ¼ 0:
ð2:17Þ
This is just the statement that, initially, there are no
G-protein activated receptors.
Next we imagine endothelial cells distributed along a
capillary of length L in some nonuniform manner.
We view this distribution as one of cell density rather
than as individual cell number. The point of view here is
the same as in quantum mechanics. We are not looking at
individual cells but rather cell density as a probability
density. Therefore, the endothelial cell density (as well as
the other quantities in Table I depend on position as well
as time. Thus, ½VðtÞ ¼ vðx; tÞ; etc. just as was done by
Levine et al. (2001b).
We now have six dependent variables s; y; c; v; r; r^ along
with three differential equations and three algebraic
equations in (2.16).
REINFORCED RANDOM WALK
It is expected that endothelial cells will move into cavities
in the extracellular matrix created by the protease they
express in response to the VEGF stimulus. The point of
view we adopt here is the same as was used by Levine and
Sleeman (1997) and Levine et al. (2001b) and is in marked
contrast to earlier works. Levine and Sleeman (1997) cites
relevant literature and the differences in approach were
spelled out in some detail.
The underlying assumptions (justified in detail by
Levine et al., 2001b) are the following:
(1) The movement of endothelial cells in tissues is not
random but depends upon the local environment of
the cells.
(2) The movement of EC in response to growth factor
is indirect. Endothelial cells will move up a protease
11
In T f < 18 h; fM moles of fibronectin will be generated by h0 endothelial cells (Yamada and Olden, 1978). In the absence of protease, h ¼ h0 where
h0 is the background concentration
of EC in a normal capillary. We assume a logistic growth of fibronectin in this case, i.e. that f t ¼ bf ð1 2 f =f M Þ for
ÐT
some b. Therefore, f M ¼ b 0 f ðtÞð1 2 f ðtÞ=f M Þ dt # f M bT f =4: The inequality will be sharp when b ¼ 4/Tf.
A MATHEMATICAL MODEL
gradient (chemotactic movement) in response to
the protease they generate when stimulated by
VEGF. This chemotactic movement is also chemokinetic, i.e. it depends upon the concentration of
protease as well as on its gradient. More precisely,
if the concentration of protease is small, but not too
small, the cells will move up the protease gradient.
However, if the protease concentration is too large,
the protease will destroy the cells (Rous and Jones,
1916). We can interpret this as saying that the
cells will avoid regions where the protease gradient
is too large or by saying that they will move down the
protease gradient in such cases.
(3) Endothelial cells will move down a fibronectin
gradient (haptotactic movement) when the fibronectin density is high and up the fibronectin gradient
when the fibronectin density is small. This is a
rough qualitative statement based on the paper by
Bowersox and Sorgente (1982), where chemotaxis of
endothelial cells in response to fibronectin was
considered.
161
It has been observed that these cells aggregate as their
food supply is consumed, i.e. as the local concentration of
cAMP increases.12 The idea of reinforced (or biased)
random walk seems to have its origins from Davis (1990).
The chemotactic sensitivity functions are phenomenological in character. For example, if the cell motion were
completely random, we take t ðc; f Þ ¼ constant: The above
equations then reduce to the one dimensional diffusion
equation and the cell density will become uniform in x
with time. If the movements depended solely upon the
gradients of c, f, a natural choice might be t ðc; f Þ ¼
const expðacÞ expð2bf Þ where a, b are positive constants.
The sensitivities vanish when t ðc; f Þ is constant, whereas,
in this case, the sensitivities are uniformly positively
correlated ða . 0Þ with protease and uniformly negatively
correlated with fibronectin ð2b , 0Þ:13
The dynamical equation (3.1) then suggests that if c, f
tend to steady state functions c1 ðxÞ; f 1 ðxÞ; the limiting
form of the cell density should be
lim hðx; tÞ ¼ Ae ðac1 ðxÞ2bf 1 ðxÞÞ
t!þ1
The cell movement equation takes the form:
›h
›
›
h
¼D
ln
h
;
›x
›x
t ðc; f Þ
›t
ð3:1Þ
which may be written in the more standard form:
tc cx þ tf f x
ht ¼ Dhxx 2 D h
t ðc; f Þ
x
¼ Dhxx 2 Dðh ðln tÞx Þx :
ð3:2Þ
The function t ðc; f Þ is called the probability transition
rate function. The ratios ›c t ðc; f Þ=t ðc; f Þ and
›f t ðc; f Þ=t ðc; f Þ are known as the chemotactic sensitivity
coefficients for protease and for fibronectin, respectively.
The Eq. (3.1) is sometimes called the continuous form
of the master equation. We shall call it the master
equation. An equation by Othmer and Stevens (1997) was
used as a model for the study of fruiting bodies such as
Myxococcus fulvus and Dictyostelium discoideum
amoeba. There t was a function of the local concentration
of cyclic adenosine monophosphate, a compound excreted
by these amoeba as they consume their local food supply.
where A is some constant of proportionality.
When the movement is chemokinetic as well as
chemotactic, the sensitivity factors will depend on c, f.
It seems reasonable to assume that cell movement will
be very sensitive to small concentrations of protease
and will be positively correlated with the enzyme
concentration. It is known that, in the presence of large
concentrations of protease, the cells will be degraded
and hence their movement will cease.
Likewise, it is reasonable to assume that cell movement
will be sensitive to low, but not too low, fibronectin
densities and insensitive to high fibronectin densities.
(If the fibronectin density is too low, the cell pseudopodia
have nothing to which to attach themselves so that they
can pull the cell along. On the other hand, if the fibronectin
density is too high in a region, then the cells cannot invade
that region. This has been documented in the literature by
Terranova et al. (1985).)
This suggests that we should take t ðc; f Þ ¼ t1 ðcÞt2 ð f Þ
since these movements should be independent.
We could take the protease sensitivity to be of the form:
t1 0 ðcÞ
g1
¼
;
t1 ðcÞ a1 þ c
12
In Levine et al. (2000), some of the mathematical properties of the solutions of the problem obtained numerically by Othmer and Stevens (1997)
were elucidated.
13
The phrase “h is positively correlated with c” is defined as follows. Consider the first order partial differential equation U t þ ½t0 ðvÞvx =t ðvÞU x ¼ 0
where U, v are functions of (x, t) (In the present context think of U ¼ h, v ¼ c and t 0 ðvÞ=tðvÞ as the chemotactic sensitivity). We say U is positively
correlated with v at a point (x, t) if the characteristic of the pde U t þ ½t 0 ðvÞvx =tðvÞU x ¼ 0 at (x, t) has positive slope if vx ðx; tÞ is positive and the
characteristic has negative slope if vx ðx; tÞ is negative. The correlation will be positive if the sensitivity t0 ðvÞ=tðvÞ is positive, negative if the sensitivity is
negative, and neutral if the sensitivity vanishes. When the sensitivity coefficient is constant, we say the correlation is uniform. The geometric meaning is
that when U is positively correlated with v, then “U will transported to the right when v is increasing and to the left when v is decreasing.” Thus if v
depends only upon x and has a single positive maximum, U will tend to aggregate at the point where the maximum occurs. Similarly, if U is negatively
correlated with v, then and v has a minimum, U will aggregate at the point where the minimum occurs.
An interesting situation arises when the sensitivity changes sign. For example, suppose t 0 ðvÞ=tðvÞ ¼ 1 2 v and vðxÞ ¼ 1 þ ð2=pÞtan21 x so that
x ¼ 0 is an inflection point for v. Then ð1 2 vÞvx is positive for x , 0 and negative for x . 0: Therefore, U will be positively correlated with v
for x , 0 and negatively correlated with v for x . 0: Thus U will aggregate at the inflection point of v, namely x ¼ 0:
162
H.A. LEVINE et al.
TABLE II Simulation parameters
Variable name
Units
26
Microns 10 m
Hours
Micro-moles/liter mM
mM
mM
mM
mM
mM
mM
Cells/liter
mM
mM h21
Position
Time
Growth factor concentration
Receptor density
Receptor–VEGF complex
Extracellular protease
Extracellular resources
Intracellular resources
Basement lamina protein
Endothelial cell (EC) density
Inhibitor concentrations ð j ¼ v; r; cÞ
Inhibitor source rates ð j ¼ v; r; cÞ
so that for the protease probability transition rate function
factor,
t1 ðcÞ ¼ Aða1 þ cÞg1
t1 ðcÞ ¼ Aða1 þ cÞg1 ða2 þ cÞ2g1 :
However, this does not convey the full thrust of item 2
above.
A more systematic way to proceed is to consider the
biology more closely. A protease sensitivity function
should have compact support contained in some interval
½0; c0 Þ vanish at the ends of this interval, and have a unique
positive maximum at some point cmax.
Then, not only will the sensitivity change sign near the
maximum of this function, the cells will tend to aggregate
near the maximum value of this function and de-aggregate
near the ends of the interval. That is, h will be positively
correlated with c for c , cmax and negatively correlated
with c for c . cmax : As the value of c approaches the end
values, 0, c0, the contribution of protease to cell movement
will become negligible if the sensitivity vanishes near the
end points.14
Because simulations will involve computation of
t1 0 ðcÞ=t1 ðcÞ; we relax the condition that t1 vanish at the
end points. We take wðcÞ to be a function of the form
described above. Then
ð3:3Þ
where a and g are positive constants chosen such that a g is
very small. Then the chemotactic sensitivity function
becomes:
0
Dimensionless variable
x
t
vðx; tÞ
rðx; tÞ
r^ ðx; tÞ
cðx; tÞ
sðx; tÞ
yðx; tÞ
f ðx; tÞ
hðx; tÞ
ij ðx; tÞ
isj ðx; tÞ
x0 ¼ x=L
t0 ¼ t=T
Vðx 0 ; t0 Þ ¼ vðx; tÞ=r0
Rðx 0 ; t0 Þ ¼ rðx; tÞ=r 0
^ 0 ; t0 Þ ¼ r^ ðx; tÞ=r 0
Rðx
Cðx 0 ; t0 Þ ¼ cðx; tÞ=r 0
Sðx 0 ; t0 Þ ¼ sðx; tÞ=s0
Yðx 0 ; t0 Þ ¼ yðx; tÞ=s0
Fðx 0 ; t0 Þ ¼ f ðx; tÞ=f M
Nðx 0 ; t0 Þ ¼ hðx; tÞ=h0
I j ðx 0 ; t0 Þ ¼ ij ðx; tÞ=r 0
I sj ðx 0 ; t0 Þ ¼ ij ðx; tÞT=r0
The choice for f is similar. Specifically, if we take cð f Þ
to be a function of the form described above, then
t2 ð f Þ ¼ ½a 0 þ c ð f Þg
where a1 is a small positive constant. However, we really
do not expect the probability rate to become infinite as c
ranges over all positive numbers. This suggests that we
take t1 as we did earlier (Levine and Sleeman, 1997;
Levine et al., 2001b) namely:
t1 ðcÞ ¼ ½a þ wðcÞg ;
Dimensioned variable
ð3:4Þ
ð3:5Þ
where a0 , g 0 are positive constants chosen such that the
product a0 g0 is very small. Then the haptotacic sensitivity
function becomes:
t2 0 ð f Þ
g 0c 0ð f Þ
¼
t2 ð f Þ a þ c ð f Þ
ð3:6Þ
Beyond this, the functional forms of f, c must be
determined experimentally although we may always
normalize them so that they take values in ½0; 1:
The larger the constant g (resp. g 0 ) is, the more concentrated about the value cmax (resp. fmax) the EC density
will be.
The specific choices we take are given in the section on
simulations.
Remark 1 This is a somewhat different philosophical approach than we took earlier (Levine and Sleeman
1997; Levine et al., 2001b). However, we believe that
this approach more accurately reflects the underlying
biology.
THE SYSTEM IN ONE DIMENSION
The system of dynamical and algebraic equations for
s; y; c; v; h; r; r^; f consists of the equations (2.16), (2.18),
(3.1), (3.3) and (3.5).
We assume initial conditions for the five differential
equations on an interval ½0; L as follows:
sðx; 0Þ ¼ s0 ; hðx; 0Þ ¼ h0 ; cðx; 0Þ ¼ 0;
0
t1 ðcÞ
gw ðcÞ
¼
:
t1 ðcÞ a þ wðcÞ
0
ð4:1Þ
f ðx; 0Þ ¼ f M ; vðx; 0Þ ¼ v0 ðxÞ;
14
The requirement that t1 have compact support can be relaxed. For example, we might consider t1 ðcÞ ¼ Ac m expð2ac n Þ rather than
t1 ðcÞ ¼ Ac m ðc0 2 cÞn : The positive constants in both forms must be determined empirically.
A MATHEMATICAL MODEL
where v0 ð·Þ will approximate a constant multiple of
unit impulse (“delta”) function. The question of
smoothness is not an issue here. The precise form
used is given in equation (7.1) below.
We write the system in dimensionless variables,
length and time scales to be selected later.
Then the system of equations to solve becomes:
!
ð t0
l1
Y ¼ Ke þ 1 2 S 2
Cðx0 ; t0 Þ þ m Cðx0 ; s0 Þ ds0 ;
n
0
the
its
we
the
ZðtÞ ¼ zðtÞ ¼ s1 SðtÞ þ sð22Þ YðtÞ;
!
ð t0
1
0 0
0 0
0
Cðx ; t Þ þ m Cðx ; s Þ ds ;
R ¼ 1 þ V 2 Vðx ; 0Þ þ
n
0
0
R^ ¼ ð1 þ l2 þ l2 VÞðVðx0 ; 0Þ 2 VÞ 2 ð1 þ l2 VÞ
"
!#
ð t0
1
Cðx0 ; t0 Þ þ m Cðx0 ; s0 Þ ds0 ;
£
n
0
r0 N
›S ;
¼ 2k2 s1 SðtÞ þ kð21Þ sð22Þ YðtÞ
0
1þZ
›t
›V
¼ 2k4 l2 RV;
›t 0
›C
^ 2 mC;
¼ n k5 ðt0 ÞRY
›t 0
›F
4
K cat l3 CF
;
¼
Fð1
2
FÞN
2
1 þ rf F
›t0 T f
›N
›
›
N
¼ D 0 N 0 ln
;
›x
›x
TðC; FÞ
›t 0
ð4:2Þ
163
TABLE III Dimensionless parameters
Initial growth factor
Vðx0 ; 0Þ ¼ vðx; 0Þ=r0
Dimensionless cell movement (diffusivity)
constant
Dimensionless protein decay rates
where mj is one of m, miv, mir, mic
Dimensionless inhibitor equilibrium constants
where nej is one of neiv ; neir ; neic
Dimensionless kinetic constants ki
where ki is one of k; kð22Þ ; k2 ; k4
Dimensionless kinetic function k5(t)
Dimensionless delay time t00
First renormalized initial receptor density
Second renormalized initial receptor density
Third renormalized initial receptor density
Normalized initial amino acid density
Normalized initial amino acid density
Normalized initial transport protein density
Normalized maximum “fibronectin” density
Dimensionless”fibronectin” time
Dimensionless Kcat
D ¼ TD=L 2
mj ¼ mj T
nej ¼ r0 nje
ki ¼ k i T
k5 ðt0 Þ ¼ T s0 k5 ðtÞ
t0 0 ¼ t0 =T
l1 ¼ r 0 =s0
l2 ¼ r 0 =K 2m
l3 ¼ r 0 =K m
s1 ¼ s0 =K 1m
sð22Þ ¼ s0 =K ð22Þ
m
r0 ¼ d0 h0 =s0 Þ
rf ¼ f M =K m
T f ¼ T f =T
K cat ¼ T K cat
micro moles per liter per hour and that the amino acid
concentration in the plasma is varying at a rate of sr ðtÞ
micro moles per liter per hour (This assumes that the
amino acids are “well mixed” while there is a spatial
distribution of growth factor). If these are written in
non dimensional variables, then with V r ðx0 ; t0 Þ ¼
Tvr ðLx0 ; Tt0 Þ=r 0 and Sr ðt0 Þ ¼ Tsr ðTt0 Þ=s0 the system (4.2)
is to be replaced by
!
ð t0
l1
0 0
0 0
0
Cðx ;t Þþ m Cðx ;s Þds
Y ¼K e þ12S2
n
0
ð t0
þ Sr ðs0 Þds0 ;
0
where TðC; FÞ ¼ tðr 0 C; f M FÞ: The sensitivity constants
ai ; bj may be redefined so that T is independent of the
scale factors r 0 ; f M . The astute reader will notice that we
cannot scale away n as the ratio C/n does not appear in the
fibronectin equation or in the cell movement equation.
This is the critical point of this paper. If n is large, the
decay in fibronectin will be very large even if v is small.
Likewise, the cell movement will be surprisingly large in
spite of the presence of only a small amount of growth
factor. Boundary conditions are needed only for the last of
the above equations. The no-flux conditions:
›
N
›
N
N 0 ln
¼ N 0 ln
›x
TðC; FÞ x0 ¼0
›x
TðC; FÞ x0 ¼1
¼0
ð4:3Þ
will suffice for our purposes.
The nondimensionalized initial conditions become:
Sðx 0 ; 0Þ ¼ 1; Nðx 0 ; 0Þ ¼ 1; Cðx 0 ; 0Þ ¼ 0;
Fðx 0 ; 0Þ ¼ 1; Vðx 0 ; 0Þ ¼ v0 ðLx 0 Þ=r 0 :
ð4:4Þ
Remark 2 It may be that growth factor is being applied to
the exterior of the basement lamina at a rate of vr ðx; tÞ
Z ¼ s1 Sþ sð22Þ Y;
R¼1þV 2Vðx0 ;0Þ2
ð t0
V r ðx0 ;s0 Þds0
0
!
ð t0
1
0 0
0 0
0
þ Cðx ;t Þþ m Cðx ;s Þds ;
n
0
0
^
R¼ð1þ
l2 þ l2 VÞ Vðx ;0Þþ
ð t0
!
V r ðx ;s Þds 2V ; ð4:5Þ
0
0
0
0
"
!#
ð t0
1
0 0
0 0
0
Cðx ;t Þþ m Cðx ;s Þds
2ð1þ l2 VÞ
;
n
0
r0 N
›S þSr ðt0 Þ;
¼ 2k2 s1 SðtÞþkð21Þ sð22Þ YðtÞ
0
1þZ
›t
›V
¼2k4 l2 RV þV r ðx0 ;t0 Þ;
›t 0
›C
^ 2 mC;
¼nk5 ðt0 ÞRY
›t 0
›F 4
K cat l3 CF
;
¼ Fð12FÞN 2
0
1þ rf F
›t T f
›N
›
›
N
¼D 0 N 0 ln
:
›x
›x
TðC;FÞ
›t 0
164
H.A. LEVINE et al.
We insert these two source terms for two disparate
reasons. In the application of this work to a
coupled system of ECM-capillary transport equations,
the source term vr ðx; tÞ will be proportional to the
concentration of VEGF molecules that have
diffused across the ECM from a remote source
(See, Levine et al. 2000; 2001b, for an illustration of
this). Also, a tumor cell that has moved away from a
remote tumor and implanted itself just inside the
capillary wall (metastasis) can serve as a source of
VEGF.
The amino acid concentration in the blood is
renewed on some continuing basis. The source
term reflects this renewal. Generally the total available
amino acid concentration will be some periodic
function of time. Tumor secreted growth factor
induces an excess production of protease by
endothelial cells from the blood amino acids that
constitutes a part of the extra burden on the body’s
resources. In particular, the larger n is, the greater is
this burden.
to t0 , we obtain to first order in e :
!
ð t0
l1
0 0
0 0
0
dy¼ 2dsþ
dcðx ;t Þþ m dcðx ;s Þds ;
n
0
!
ð t0
1
0 0
0 0
0
dr ¼ dv2 dv0 þ dcðx ;t Þþ m dcðx ;s Þds ;
n
0
dr^¼ ð1þ l2 Þðdv0 2 dvÞ
"
!#
ð t0
1
0 0
0 0
0
2
dcðx ;t Þþ m dcðx ;s Þds
;
n
0
ðdsÞt0 ¼ ½2k2 s1 dsðtÞþkð21Þ sð22Þ dyðtÞr0 =
£ð1þ s1 þK e sð22Þ ÞðdvÞt0
¼ 2k4 l2 dv;ðdcÞt0 ¼nk5 ðt0 ÞK e dr^2 md
c;ðdf Þt0
¼
K cat l3 dc
24
df þ
;
Tf
1þ rf
› ›C Tð0;1ÞðdcÞx 2 ›F Tð0;1Þðdf Þx
ðdhÞt0 ¼ D ðdhÞxx 2
:
›x
Tð0;1Þ
INSTABILITY ANALYSIS
The system (4.2) together with Eqs. (4.3) and (4.4) can
be viewed as a dynamical system in which we are
perturbing the rest state
^ V; C; F; Nl
Re ¼ kS; Y; R; R;
Thus, dv converges uniformly and exponentially
rapidly to zero. If we assume that, as t0 ! þ1; the
solution of Eq. (5.2) converges to a steady state, then,
suppressing the argument x0
lim kds; dy; dr; dr;^ dv; dc; df ; dhl
¼ k1; K e ; 1; 0; 0; 0; 1; 1l
t0 !1
by perturbing Vðx0 ; 0Þ from zero. As is well known,
theorems that claim stability of rest states from
statements of their linearized stability, are rare and, in
many cases involving nonlinear partial differential
equations, are nonexistent. However, the converse is
true, namely, if the linearized problem is unstable, then
so is the nonlinear problem.
Suppose, as is the case biologically, that Vðx0 ; 0Þ ¼
edv0 ðx0 Þ where e . 0 is small. Then set
^ V; C; F; Nl ¼ k1 2 eds; K e 2 edy;
kS; Y; R; R;
¼ kdse ; dye ; dr e ; re ; dve ; dce ; df e ; dhe l;
ds e ¼
ð1
l1
dce ðxÞ þ m dcðx0 ; s0 Þ ds0 :
nð1 þ K e Þ
0
In order that the integral on the right converge, we
must have dce ðx0 Þ ¼ 0: Thus, for each x0 ,
0
ð5:1Þ
where dg denotes a small perturbation in the quantity g.
Using gt0 to denote partial differentiation with respect
ð5:3Þ
dve ¼ 0 and dye ¼ K e dse : From the first of Eq. (5.2)
we see that
ð1
1 þ edr; edr;^ edv; edc;
1 2 edf ; 1 þ edhl
ð5:2Þ
dcðx0 ; s0 Þ ds0 ¼
ndse ðx0 Þ
:
mðK e þ 1Þ
ð5:4Þ
Ð 1Since0 dc0 e ¼0 0; we must also have dr^e ¼ 0: Therefore,
0 dcðx ; s Þ ds ¼ nð1 þ l2 Þdv0 =m: This gives us an
indication of how much protease we can expect from the
system for small concentrations of growth factor. Setting
ðdf Þt ¼ 0 and using dce ¼ 0 again, we see that df e ¼ 0:
A MATHEMATICAL MODEL
Consequently, dhe;xx ¼ 0: Therefore, using the boundary
conditions we conclude that dhe ; 0:
Summing up, we have, together with Eq. (5.4)
dRe ¼ kdse ; dye ; dre ; dr^e ; dve ; dce ; df e ; dhe l
¼ dv0 ð1 þ l2 Þ
l1
K e l1
l2
;
;
; 0; 0; 0; 0; 0 : ð5:5Þ
K e þ 1 K e þ 1 1 þ l2
(The apparent increase in available receptors is an
artifact of the Michaelis – Menten assumption in Eq. (2.14).
We assumed at the outset that r0 receptors were free and
r 0 v0 =K 2m were bound. When all the growth factor is gone
from the system, the number of free receptors returns to its
expected value, r 0 þ r 0 v0 =K 2m (or, in this case, 1 þ edr e )
which is the concentration of free receptors together with
the concentration of receptors that are initially bound up
with the growth factor bolus).
This means that under small perturbations, the
linearized system (5.2) carries the rest state Re to a new
rest state Re þ edRe and hence cannot be stable.
We illustrate this instability in the computations below.
165
We begin with the case of growth factor inhibition.
Thinking of the concentration of each species as both space
and time dependent, the total concentration of growth
factor, vtot, in the system consists of the concentration of
active molecules (va), the concentration of inhibited
molecules vi, the concentration of molecules bound to
receptors m and the concentration of molecules that have
been degraded v
a : If we assume that the rate of supply of
growth factor is vr ðx; tÞ then we have the following
dynamics and conservation laws:
›va
¼ kð23Þ mðtÞ 2 k3 rðtÞva ðtÞ 2 vsa ðx; tÞ;
›t
›m
¼ k3 rðtÞva ðtÞ 2 ðkð23Þ þ k4 ÞmðtÞ;
›t
›r
¼ k5 ðtÞ^rðtÞyðtÞ þ kð23Þ mðtÞ 2 k3 rðtÞva ðtÞ;
›t
›r^
¼ k4 mðtÞ 2 k5 ðtÞ^rðtÞyðtÞ;
›t
ð6:4Þ
›v
a
›vtot
¼ k4 mðtÞ;
¼ vr ðx; tÞ;
›t
›t
vtot ¼ va þ vi þ m þ v
a ;
vi ¼ nve va iv
INHIBITION
One would like to inhibit the production of protease with
some sort of inhibitor. There are, as one sees from the
mechanism described in the appendix below, several
points at which inhibition would be effective.
For example, in the overall mechanisms (2.5) –(2.8),
one might try to inhibit protease production with a
protease inhibitor, or growth factor with a growth factor
inhibitor, or try to block receptor function with a receptor
inhibitor. (Blocking angiogenesis by interfering with
various steps in the signaling pathway is under
consideration at the experimental level. See, Eckhard
(1999) for a nice illustration of some of the steps that are
being selected as possible targets.)
In order to analyze such statements, we argue as
follows:
If I v ; I c ; I r are inhibitor molecules, consider the
equilibria:
Iv þ V A Y V I ;
ð6:1Þ
I r þ RA Y RI
ð6:2Þ
and
I c þ CA Y CI ;
ð6:3Þ
where the subscripts A, I refer to the active and inert forms
of the molecular species to which the subscript is attached.
Let nev ; nec ; ner be the equilibrium constants for each of
these reactions.
where iv is the concentration of inhibitor and where vsa is a
sink term to be determined that describes the effect of the
inhibitor on the growth rate of active receptors. In order to
determine the form of vsa , we take the time derivative of the
seventh equation after using the eighth equation to
eliminate vi and the first equation to eliminate ›t va to obtain
vr ¼ ð1 þ nve iv Þvsa þ va nve ›t iv
þ nve iv ½kð23Þ mðtÞ 2 k3 rðtÞva ðtÞ:
This leads us to:
›
½va ð1 þ nve iv Þ ¼ kð23Þ mðtÞ 2 k3 rðtÞva ðtÞ þ vr ðx; tÞ:
›t
It is convenient to define v ¼ va þ vi ¼ ð1 þ n ev iv Þva
as the free growth factor, i.e. the concentration of growth
factor that is neither receptor bound nor destroyed. Then
the differentiated form of the equations which replace the
last two equations in (2.13):
›
ð^r þ m þ rÞ ¼ 0
›t
ð6:5Þ
›
ð^r þ m þ vÞ ¼ 2k5 r^y þ vr :
›t
These equations are of the same form as the
differentiated form of the last two equations in (2.13)
with the exception that now we have included the
source term for growth factor in the second equation.
166
H.A. LEVINE et al.
Now the second equation in (2.14) takes the form
mðtÞ ¼
rðtÞva ðtÞ
:
K 2m
laws:
ð6:6Þ
We will also need an equation which describes the
time dynamics for the inhibitor:
›iv
¼ isv ðtÞ 2 miv iv :
›t
ð6:7Þ
where, from the point of view of the patient, the half
life, ln 2=miv ; should be large. The initial condition for
Eq. (6.7) may be taken to be iv ðx; 0Þ ¼ i0 ðxÞ (If the
inhibitor is introduced intravenously, we may take
i0 ¼ 0 and isv to be a constant).
We list here only the dimensionless form of the
equations in Eq. (4.5) which must be changed to reflect
the altered dynamics:
R^ ¼
1þ
l2 ð1 þ Vðx0 ; t0 ÞÞ
1 þ nev I v ðx 0 ; t0 Þ
Vðx 0 ; 0Þ þ
£
ð t0
›r a
¼ k5 ðtÞ^rðtÞyðtÞ þ kð23Þ mðtÞ 2 k3 r a ðtÞvðtÞ 2 r sa ;
›t
›m
¼ k3 r a ðtÞvðtÞ 2 ðkð23Þ þ k4 ÞmðtÞ;
›t
›v
¼ kð23Þ mðtÞ 2 k3 r a ðtÞvðtÞ;
ð6:9Þ
›t
›r^
¼ k4 mðtÞ 2 k5 ðtÞ^rðtÞyðtÞ;
›t
r total ¼ r a þ r i þ m þ r^ ¼ r 0 þ m0
r i ¼ ner r a ir
where now r sa is a sink for activated receptors. It is clear that
›
ð^r þ m þ r a Þ ¼ 2r sa ;
›t
›
ð^r þ m þ vÞ ¼ 2k5 r^y þ vr :
›t
ð6:10Þ
From the first of these and the fifth of Eq. (6.9), we
see that we have no choice but to take
!
r sa ¼ ›t r i :
V r ðx 0 ; s0 Þds 0 2 Vðx 0 ; t 0 Þ ;
0
2 1þ
l2 Vðx ; t Þ
Þ
1 þ nev I v ðx0 ; t0 Þ
0
With this choice, we are once again led to:
0
"
!#
ð t0
1
0 0
0 0
0
Cðx ; t Þ þ m Cðx ; s Þ ds
£
;
n
0
›
ð^r þ m þ rÞ ¼ 0;
›t
ð6:8Þ
›V
l2 V
þ V r ðx0 ; t0 Þ;
¼ 2k 4 R
1 þ nve I v ðx0 ; t0 Þ
›t 0
›I v
¼ I sv ðtÞ 2 miv I v :
›t 0
(The equation for R is unchanged from that
Eq. (4.5).) The source term isv has been replaced by
its non dimensional form I sv ðt0 Þ ; T isv ðt0 Þ=r 0 ; iv by
I v ¼ iv =r 0 ; nve ¼ r 0 nve and miv ¼ T miv .
The astute reader will note that the inhibition of V is
expressed by the replacement of l2 by l2 =ð1 þ
nev I v ðx0 ; t0 ÞÞ: As we let nev increase without bound, i.e.
as we drive the equilibrium to the right, this coefficient
will tend to zero. This will drive R to unity and R̂ to
zero and there will tend to be very few activated
receptors to convert intracellular resources into
protease.
We turn next to Eq. (6.2), the case of receptor inhibition.
Here the situation is somewhat similar to the first case.
We have r total ¼ r a þ r i þ m þ r^ ¼ r 0 þ m0 as the concentration of receptors available a very short time after
the reaction has begun (Again, each variable is potentially
a function of position and time). The chemistry dictates
the following dynamical equations and conservation
where now r ¼ r a þ r i ¼ ð1 þ ner ir Þr a denotes the concentration of receptors per cell at time t which are not
bound to growth factor nor part of an activated receptor
complex. The second equation in (2.14) takes the form
mðtÞ ¼
r a ðtÞvðtÞ
:
K 2m
ð6:11Þ
The altered dynamical equations are of the same form
as given in Eq. (6.8) namely
l2 ð1 þ Vðx0 ; t0 ÞÞ
R^ ¼ 1 þ
1 þ ner I r ðx0 ; t0 Þ
0
Vðx ; 0Þ þ
£
ð t0
!
0
0
0
0
0
V r ðx ; s Þds 2 Vðx ; t Þ ;
0
2 1þ
l2 Vðx0 ; t0 Þ
1 þ ner I r ðx0 ; t0 Þ
"
!#
ð t0
1
0 0
0 0
0
Cðx ; t Þ þ m Cðx ; s Þ ds
£
;
n
0
›V
l2 V
þ V r ðx0 ; t0 Þ;
¼ 2k 4 R
1 þ nre I r ðx0 ; t0 Þ
›t 0
›I r
¼ I sr ðtÞ 2 mir I r :
›t 0
ð6:12Þ
A MATHEMATICAL MODEL
The source term isr has been replaced by its non
dimensional form I sr ðt0 Þ ; T isr ðtÞ=r 0 ; ir by I r ¼ ir =r 0 ;
nre ¼ r0 nre and mir ¼ T mir as before.
Thus the dynamics of inhibition of growth factor
(Eq. (6.8)) or the inhibition of receptor activation
(Eq. (6.12)) have precisely the same form. This model
predicts that equal inhibitor equilibrium constants and
equal bolus concentrations or source rates of either type of
inhibitor will result in equal inhibition of fibronectin decay
and aggregation of endothelial cells.
Remark 3 If one introduces both inhibitor types the
resultant equations are modified to the extent that l2 in
Eq. (4.5) is replaced by
ð1 þ
l2
r
0
0
ne I r ðx ; t ÞÞð1
þ nev I v ðx0 ; t0 ÞÞ
wherever it appears. Therefore, even if the equilibria for
both types of inhibition are relatively modest, i.e. the
constants nve ; ner are not inordinately large, the combined
effect of two such inhibitors is greater by a factor of one of
them over the other than either one alone (This result is not
unexpected. It is dictated by the kinetics. If both V and R
are inhibited, then the concentration of ½V A RA is very
nearly proportional to the product of the concentrations
of each active species by our version of the Michaelis–
Menten hypothesis).
When Eq. (6.3) is the mechanism, the situation is much
easier to describe: First, the concentration of protease is
replaced by the concentration of active protease in the
fibronectin and EC movement equations. Then
½C ¼ ½C A þ ½C I þ ½C A F
¼ ½C A þ ½C I þ ½C A ½F=K m ;
ð6:13Þ
167
the first seven equations in (4.5) are unchanged while
the last two and the cell movement equation become:
›I c
¼ 2mic I c þ I sc ðt0 Þ;
›t 0
›F
4
¼ Fð1 2 FÞN 2 K cat l3 C a F;
›t0 T f
ð6:16Þ
›N
›
›
N
¼ D 0 N 0 ln :
›x
›x
›t 0
TðCa ; FÞ
A final observation: it is straightforward to modify
either Eq. (4.2) or (4.5) in the case that one has one or
more inhibitors, one which acts against growth factor, a
second which acts against receptor signaling, and a third
which acts against protease.
THE NUMERICAL SIMULATIONS
Below we present some simulations using an initial bolus
of growth factor rather than a source. We take
v0 ðxÞ
¼
8
d
< V 0 N½12cosð2pðx2xl Þ=ðxr 2xl ÞÞ if xl #x#xr
:0
if 0#x,xl or xr ,x#L
ð7:1Þ
where N ¼NðdÞ is a normalizing constant chosen so that
ð xr
v0 ðxÞdx ¼V 0 :
xl
½CI ¼ nce ½I c ½C A :
The total concentration of enzyme available for
protease degradation is ½C 2 ½CI :
We see that
½C A ¼
1þ
½C
þ ½F=K m
nce ½I c ð6:14Þ
which must be small in order to inhibit the onset of
angiogenesis. This will be the case if nce is very large and
n is not too large in the case that the inhibitor
concentration is modest. Unfortunately, n is large, and
for at least one inhibitor, plasminogen derived angiostatin
which is an inhibitor of tPA, the equilibrium constant is of
the order of one (mM)21 and hence is not very large.
Suppressing ðx0 ; t0 Þ and passing to dimensionless
variables,
C ¼ C a þ Ci þ rf Ca F;
ð6:15Þ
C i ¼ nec C a I c ; C a ; C 2 Ci :
This choice, for large d corresponds roughly to a
d-function bolus of magnitude V0. Since the amplification
factor n is not known (and is also growth factor dependent)
we have taken it to be a constant, for lack of better
information at the current writing.
We used the values in Table IV for the various
parameters and constants: in the above table, the constants
K 1m ; K 1cat are taken from Heaton and Gelehrter (1977). The
constants k5, t0 are based on the estimated EC response
time to growth factor (Unemori et al., 1992). The choices
2
cðFÞ ¼ 4Fð1 2 FÞ and fðcÞ ¼ Ace 2jc were taken in the
probability transition function. (Here A is the reciprocal of
2
the maximum value of ce 2jc ).
In the actual simulations below we have reduced the
constant S0 by a factor of 105 in order to illustrate how the
growth factor draws on the external resources via this
transfer mechanism.
(1) In the first set of simulations, illustrated by the eight
panels in Figs. 2 and 3, we fix the initial concentration of growth factor. There are two input sources
168
H.A. LEVINE et al.
TABLE IV Parameter values
References
K m ¼ 0:7813 mM
K 1m ¼ 2:93ð1023 ÞmM
K 2m ¼ 1:4286ð102 ÞmM
k ¼ 0:6667ð1021 Þh21
r0 ¼ 1:0 mM
p0 ¼ 1:0 mM
mr ¼ 0:01 £ m
mc ¼ 0:01 £ m
nre ¼ 5:0ð106 ÞmM
nce ¼ 5:0ð106 ÞmM
iv ðx; 0Þ ¼ 5:0ð1024 ÞmM
ic ðx; 0Þ ¼ 5:0ð1024 ÞmM
isr ðx; tÞ ¼ 0:0
isc ðx; tÞ ¼ 0:0
T ¼ 1:0 h
D ¼ 3:6ð1025 Þmm2 h21
xl ¼ 0:0
f M ¼ 1:0ð1022 ÞmM
a1 ¼ 1:0ð1023 Þ
g1 ¼ 1:2
n ¼ 2ð103 Þ
V 0 ¼ 2:5ð1024 ÞmM
K cat ¼ 1:484ð10Þh21
k2 ¼ K 1cat ¼ 9:42ð1028 Þh21
k4 ¼ 1:04286ð104 Þh21
t0 ¼ 1:5ð10Þh
s0 ¼ 2:442ð103 ÞmM
m ¼ 4:56 h21
mv ¼ 0:01 £ m
nve ¼ 5:0ð106 ÞmM
ir ðx; 0Þ ¼ 5:0ð1024 ÞmM
Fields et al. (1990)
Heaton and Gelehrter (1977)
Kendall et al. (1999)
See footnotes
Terman et al. (1992); Waltenberger et al. (1994) for r0, Engelen et al. (2000) for s0
Assumed that p0 < r0 : Boffa et al. (1998) for m
Simulated value
Simulated value
Simulated value
Simulated value
Simulated value
Simulated value
isv ðx; tÞ ¼ 0:0
T f ¼ 1:8ð10Þh
L ¼ 100m ¼ 0:1 cm
xr ¼ 100m
b1 ¼ 1:0ð1023 Þ
g2 ¼ 1:2
d ¼ 30
Time scale. Orme and Chaplain (1996); Yamada and Olden (1978) for Tf
Sherrat and Murray (1990) for D. Length scale
Terranova et al. (1985) for fM. See foonotes for CM
Simulated value
Simulated value
Simulated value
Simulated value
which regulate the EC-fibronectin response. The first
is the quantity of growth factor in the bolus while
the second is the quantity of externally supplied
resources reflected in the magnitude of S0. This is
quite a large quantity, and for the levels of growth
factor which are found in tissue samples, is far more
than is needed to drive the computations. Therefore,
in the figures below, we have reduced the number
S0 by a factor of 105. The second regulator of EC
response is the magnitude of the growth factor bolus.
The computations illustrate the instability discussed above. As one can see from the figures,
although the growth factor decays very rapidly due to
the large influx of extracellular resources, its effects
FIGURE 2 Extra- and intracellular resource, receptor and activated receptor time courses without inhibitor.
A MATHEMATICAL MODEL
169
FIGURE 3 Growth factor, protease, fibronectin and endothelial cell time courses without inhibitors.
FIGURE 4 Extra- and intracellular resources, receptor and activated receptor time courses with growth factor or receptor inhibitor ðtv ¼ 0:0Þ:
170
H.A. LEVINE et al.
FIGURE 5 Growth factor, protease, fibronectin and endothelial cell time courses with growth factor or receptor inhibitor ðtv ¼ 0:0Þ:
FIGURE 6 Extra- and intracellular resources, receptor and activated receptor time courses with growth factor or receptor inhibitor ðtv ¼ 50:0Þ:
A MATHEMATICAL MODEL
FIGURE 7 Growth factor, protease, fibronectin and endothelial cell time courses with growth factor or receptor inhibitor ðtv ¼ 50:0Þ:
FIGURE 8 Extra- and intracellular resources, receptor and activated receptor time courses with protease inhibitor ðtc ¼ 50:0Þ:
171
172
H.A. LEVINE et al.
are felt after several hours in the form of a protease
“bolus”. This bolus rapidly disappears although the
fibronectin and EC profiles “remember” it. Notice
that the EC concentrations in panel 8 clearly follow
the fibronectin concentrations after 100 h (when the
protease has nearly all decayed) as the theory dictates.
Observe also that the fibronectin density vanishes
over an interval of approximately 6– 8 microns, about
the diameter of a capillary. This opening is “lined”
with EC in the sense that the EC density is highest
along the edges of the capillary opening and vanishes
near the center of the capillary opening.
(2) With the above bolus of growth factor, a uniform
bolus of growth factor inhibitor (or, as remarked
above, receptor inhibitor) of concentration of
5:0ð1024 ÞmM is introduced in the blood stream. The
results are illustrated in the second set of eight panels.
It is assumed that this inhibitor is very effective ðne <
106 mMÞ and that it has a long half life (We have taken
this to be 100 times longer than that of the protease).
In Figs. 4 and 5, the inhibitor bolus was introduced at
the same time as the growth factor bolus was
introduced. Notice that although the channel opening
does not close, it does become more sharply defined.
It is also narrower than had the inhibitor not been
introduced.
The effect of the inhibitor is to delay action of the
growth factor. As the inhibitor decays, the equilibrium
between the inhibited and uninhibited forms of the
receptor (or growth factor) shifts to the left releasing
more of the uninhibited receptor or growth factor.
If the decay constant for inhibitor is set to zero, then
the system shifts to a new state in which l2 is replaced
by l2 =ð1 þ ne I 0 Þ where ne is one of the two
equilibrium constants and I0 is the magnitude of the
inhibitor bolus.
In Figs. 6 and 7, the inhibitor bolus was introduced
50 h after the growth factor bolus was introduced.
Notice that while the protease density is reduced
considerably, the channel does not close at those
points where it was completely open (i.e. where
Fðx; tÞ ¼ 0). Notice the attempt by the endothelial cell
density to spread out at and slightly after 50 h. The
channel opening does not (and, in this model, cannot)
narrow in this case.
If we had included a diffusion term in the
fibronectin equation, then the channel opening
would close, although, since the diffusion constant
for fibronectin is very small, this would not be
noticeable here (Levine and Sleeman, 1997).
(3) Finally, a protease inhibitor is introduced in lieu of the
other two inhibitor types. Here again, it is assumed
that this inhibitor is long lived and very effective. The
inhibitor bolus was introduced fifty hours after the
growth factor bolus. (Introducing the growth factor
bolus at time zero just means that we will have
FIGURE 9 Growth factor, protease, fibronectin and endothelial cell time courses with protease inhibitor ðtc ¼ 50:0Þ:
A MATHEMATICAL MODEL
a decayed value of inhibitor at the time of onset of
protease production. This again involves a simple
rescaling of l2 at that time) (Figs. 8 and 9).
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of cell surface and blood”, Nature, 179–184.
APPENDIX
The process, as currently understood, by which growth
factor signals endothelial cells to generate protease may
be outlined as follows:
(i) A growth factor molecule binds to a cell receptor to
form a receptor complex [RV ]. The receptor
complex is activated by this binding. The activated
receptor modifies (phosphorylates) portions of
itself such that it is now attractive for the binding
of an adapter protein that in turn binds a GDP/GTP
exchange factor. This exchange factor forms a
complex with a monomeric G-protein with the
resulting G-protein– RV complex, G0 .
This activated complex is then taken into the
interior of the cell and follows one of two
pathways.
(ii) The activated G complex, G0 , can break down in
three steps to yield degraded growth factor and
receptor products as well as the original molecule
of G-protein.
(iii) Before following IIa, the activated complex
activates a series of enzymes in several steps.
First it activates Raf (MAP kinase – kinase –kinase).
174
(iv)
(v)
(vi)
(vii)
(viii)
(ix)
H.A. LEVINE et al.
The activated Raf then activates Mek (MAP
kinase –kinase). Activated Mek then follows one
of two paths.
Mek forms a non-competitive feedback loop by
activating phosphatase in the cytoplasm which, in
its turn, acts on activated Raf to return it to the
inactive state.
Mek goes on to interact with MAP kinase to produce
activated MAP kinase.
Activated MAP kinase activates transcription factor
either in the cytoplasm or by first moving to the
nucleus and activating transcription factors that are
resident in the nucleus.
The activated transcription factor activates the
DNA to form an activated DNA-transcription factor
complex.
RNA polymerase is activated by the transcription
factor complex to synthesize messenger RNA
(mRNA) encoding protease. The mRNA is
transported to the cytoplasm where the ribosome
translates it to assemble the protease from the
cytoplasmic amino acids.
RNA polymerase is also activated by this complex
to synthesize messenger RNA (mRNA) encoding
receptor protein. This mRNA is transported to the
cytoplasm where ribosome translates it to
reassemble a receptor of the original type from the
cytoplasmic amino acids.
The process (viii), will in general, be repeated several
times in succession resulting in the production of several
molecules of protease being synthesized for each molecule
of R that reappears on the cell surface (That is, in the time
that a receptor is degraded, resynthesizes via (ix) and
exported to the surface of the cell, several molecules of
protease will have been synthesized) (Table V).
The result of both of (viii) and (ix), is a depletion of the
cell amino acids. These cytoplasmic amino acids are
replaced by amino acids from the ECM or the blood plasma
as outlined above, concurrently with the assembly of the
protease molecules. We begin with item (i) in the outline
above.
k1
k2
V þ R Y RV; RV ! ðRVÞ0 ;
ð8:1Þ
kð21Þ
which summarizes the attachment and activation of the
receptor R. Then the G-protein binds with this activated
complex
k3
ðRVÞ0 þ G Y E;
k23
k4
E Y G0 þ ðRVÞ0 ;
ð8:2Þ
k24
where G0 is the activated G-protein complex with (RV)0 .
During this process, (RV)0 is being invaginated by the cell.
During this process two events occur. For (ii),
k5
k6
k7
ðRVÞ0 !ðRVÞ00 ; G0 !G; ðRVÞ00 !V* þ R*
ð8:3Þ
where V*, R* are the resultant products of V, R degradation.
For (iii), along the scaffold, G0 catalyzes K vis:
k9
k10
ð8:4Þ
k12
ð8:5Þ
G0 þ K Y G0 K; G0 K ! G0 þ K 0 :
k29
Then K0 catalyzes L vis:
k11
K 0 þ L Y K 0 L; K 0 L ! L0 þ K 0
k211
At this point, one of two further events occur, (iv),
which is a non-competitive feedback loop:
k13
k14
L0 þ Z Y L0 Z; L0 Z ! L0 þ Z 0
ð8:6Þ
k213
k15
k16
Z0 þ K Y Z0K0; Z0K0 ! K þ Z0:
ð8:7Þ
k215
Or, for (v), L0 goes on to activate M:
k17
k18
L0 þ M Y L0 M; L0 M ! L0 þ M 0
ð8:8Þ
k217
TABLE V Nomenclature for reaction mechanisms
Species
Growth factor
Cell receptor
Amino acids
Sugars/bases in mRNA
Degraded VEGF residues
Degraded receptor residues
Growth factor receptor complex
Activated growth factor receptor
complex
G-protein
Activated G-protein
Activated intermediate complex
Raf
Activated Raf
Mek
Activated Mek
Phosphatase
Activated phosphatase
MAP kinase
Activated MAP kinase
Transcription factor
Activated transcription factor
Activated transcription factor
DNA
Activated DNA
Ribosome
DNA
Activated DNA
Messenger RNA
Messenger RNA
RNA polymerase
Protease
Protease
“Nascent” cell receptor
Notation
Source/location
V
R
Y
B
V*
R*
RV
(RV)0
Extracellular matrix
Trans-membrane
Cytoplasm
Cytoplasm
Cytoplasm
Cytoplasm
Trans-membrane
Trans-membrane
G
G0
G0 (RV)0
K
K0
L
L0
Z
Z0
M
M0
Tr
Tr0
Tr00
Dn1
Dn10
Rb
Dn2
Dn20
Rn1, Rn2
Rn10 , Rn20
P
C0
C
R0
Cell cytoplasm
Cell cytoplasm
Cell cytoplasm
Cytoplasm scaffold
Cytoplasm scaffold
Cytoplasm scaffold
Cytoplasm scaffold
Cytoplasm
Cytoplasm
Cytoplasm scaffold
Cytoplasm scaffold
Cytoplasm
Cytoplasm
Nucleus
Nucleus
Nucleus
Cytoplasm
Nucleus
Nucleus
Cytoplasm
Nucleus
Nucleus
Cytoplasm
Extra cellular matrix
Cytoplasm
A MATHEMATICAL MODEL
which in turn activates transcription factor Tr:
k19
175
k29
k30
Dn2 0 þ P Y Dn2 0 P; Dn2 0 P!Dn2 0 þ Rn2 0 ;
k229
k20
T r þ M 0 Y T r M 0 ; T r M 0 !T r 0 þ M 0 :
ð8:9Þ
If it is not already resident in the nucleus, the
transcription factor makes its way to the cell nucleus via
kcn
20
T r 0 ! T r 00
k30
Dn2 0 !Dn2
k219
ð8:10Þ
ð8:17Þ
where again the last step indicates the return of the
activated DNA to the inactive state. The mRNA is modified
and then transported back to the cytoplasm via
k30nc
where “cn” means “cytoplasm – nucleus”. The path again
splits, this time in the nucleus, via paths (viii) and (ix).
The activated transcription factor may activate the DNA to
produce protease via
Rn2 0 ! Rn2 :
k31
Rn2 þ Rb Y Rn2 Rb;
k231
ð8:19Þ
k32
00
k22
ð8:11Þ
00
0
T r Dn1 ! T r þ Dn1 :
k23
k24
Dn1 0 þ P Y Dn1 0 P; Dn1 0 P ! Dn1 0 þ Rn0 ;
k24 0
k30rna
ð8:12Þ
where the last step indicates the return of the activated
DNA to the inactive state.
The mRNA is modified and then transported to the
cytoplasm via
k24nc
Rn1 0 ! Rn1 :
where Y denotes the concentration of the available amino
acids in the cytoplasm. It is assumed here that the ribosome
reads the mRNA just once before it is degraded via:
Rn2 ! B
k223
Dn1 0 ! Dn1
0
Y þ Rn2 Rb ! Rn2 þ Rb þ R ;
k21
T r 00 þ Dn1 Y T r 00 Dn1 ;
k221
ð8:18Þ
into the bases and sugars of which it is comprised.
The protease C0 is moved to the cell exterior by a
transfer mechanism that involves lipid channels and lipid
vesicles:
k33
P t þ C 0 Y Pt C 0 ;
k233
ð8:13Þ
ð8:20Þ
ð8:21Þ
0 k34
Pt C ! Pt þ C:
(This is actually a several step process, each with a
rate constant, some of which happen simultaneously).
The ribosome then translates the mRNA into
protease:
The receptor protein R0 moves to the lipid bilayer via
k35
R0 ! R:
ð8:22Þ
k25
Rn1 þ Rb Y Rn1 Rb ;
k225
ð8:14Þ
k26
Y þ Rn1 0 Rb ! Rn1 þ Rb þ nC0
where Y denotes the concentration of the available
amino acids in the cytoplasm. The factor n is included
here to represent the number of times the ribosome
reads the mRNA before it is degraded via:
k24rna
Rn1 ! B
ð8:15Þ
into the bases and sugars of which it is comprised.
The other part of the path, (ix), reproduces the cell
receptor:
k27
k28
T r 00 þ Dn2 Y T r 00 Dn2 ; T r 00 Dn2 ! T r 00 þ Dn2 0 :
k227
ð8:16Þ
In addition to these we also need to write down
mechanisms by which the proteases and receptors are
transfered back to the extracellular matrix. In so far as
the actual protein translation steps are concerned,
Eqs. (8.14), (8.15), (8.19) and (8.20) are only gross
simplifications of these complicated events.
Clearly an attempt to write down the kinetic
equations for the system of reactions (8.1) –(8.22) without
some assurance of the availability of the kinetic constants
for these equations and some assurance that the rate
determining steps are all included in the above sequence
would be folly. On the other hand, there is some
educational value in recording the chemical equations and
the pathways. Such a long chain of chemical events
certainly demands some realistic simplification such as
described above.