We will now consider what are the essential elements for
modeling cancers. The first step is to re-establish the goals of a model and
then its structure. Finally we will lead into the interrelationship between a
model and the data which is used to justify it.This work is detailed in a recent White Paper.
Many authors have developed models concerning pathways and
also cancer. The books by Klipp et al and that of Szlassi et al are excellent
overviews of the area with significant detail. The Klipp et al book is a truly
superb discussion regarding pathways and modeling alternatives. The books by
Bellomo et al and Wang are directed specifically at cancer modeling but
unfortunately they lack adequate pathway dynamics to be of substantial use. Yet
they are the only books available within the focused area.
At the core, we want a model which reflects the following
qualities:
1. Based Upon Reality: The model must at its core be based
upon the known reality. It must conform with what we currently know and
understand. Namely it must reflect in its core the elements which we consider
critical and the temporal and spatial dynamics of those elements. The model
must be based upon a tempero-spatial system of measurable quantities ;linked in
some kinetic manner using reasonably well understood processes.
2. Predictability: Any modeling must, if it is to have any
credibility, have the ability to predict, to say what will happen, and then to
have that prediction validated. Although the ability may be statistical in
nature the statistical confidence must be justifiable. We know all too well
that many things are correlated, yet not causal, and not predictable.
3. Measurable: One must be able to measure and then predict
the quantities which make up the model. Many of the modeling systems include
proteins but they react in some zero-one format. We know in reality that we
have concentrations, or better yet specific numbers of proteins, produced in a
cell. Yet we cannot yet measure the number of each of these proteins. We all
too often can at best measure their presence or absence. However, is it not the
case that it is the excess or the low density of some set of proteins which
shift reactions, and that reactions are often concentration dependent.
4. Modellable: We want a system which can be modeled. It
must reflect the measurable quantities in space and time and the
tempero-spatial dynamics of them, using techniques that we can then use for
prediction and validation.
In this paper we examine and analyze several models of
cancer. Specifically we look at intracellular, extracellular and full body
models. We attempt to establish a linkage between all of them. Many researchers
have looked at the gene level, the pathway level and the gross flow of cancer
cell level, namely whole body. Connecting them has been complex to say the least.
But herein we look at the pathway level and a whole body
level and demonstrate the nexus, physically, and from this we argue that one
can construct both prognostic tools as well as methodologies to deal with
metastasis.
The following graphic lays out the flow of development and
its implications as we detail them herein.
The key question we ask is just what is it we are modeling
in cancer cell dynamics. Let us consider some options:
This type of model focuses on the genes, and their behavior.
It is basically one where we examine the gene type and its product.
This type of model falls in several subclasses. All begin
with protein pathways and the “dynamics” of such pathways. But we have two
major subclasses; protein measures and temporal measures. By the former we mean
that we can look at the proteins as being on or off, there or not there, or at
the other extreme looking at the total number of proteins of a specific type
generated and present at a specific time. By the latter, namely the temporal
state, we can look at the proteins in some static sense, namely there or not
there at some average snapshot instance, or we can look at the details over
time, the detailed dynamics. In all cases we look at the intracellular dynamics
only.
Let us consider the two approaches.
i. On-Off: In this approach the intracellular relationships
are depicted as activators or inhibitors, namely if present they allow or block
an element in a pathway. PTEN is a typical example, if present it blocks Akt,
if absent it allows Akt to proceed and enter mitosis. p53 is another example
for if present we have apoptosis and if absent we fail to have apoptosis. These
are simplistic views. This is a highly simplistic view but it does align with
the understanding available say with limited microarray techniques. This is an
example of the data collection defining what the model is or should be.
ii. Density: This is a more complex model and it does
reflect what we would see as reality. The underlying assumptions here are:
a. Genes are continually producing proteins via
transcription and translation.
b. Transcription and translation are affected at most by
proteins from other genes acting as repressors or activators. There are no
other elements affecting the process of transcription and translation. Not that
this precludes any miRNA, methylation, or other secondary factors. We shall
consider them later. In fact they may often be the controlling factors.
c. The kinetics of protein production can be determined.
Namely we know the rate at which transcription and translation occur in a
normal cell or even in a variant. That is we know that the production rate of
proteins can be given by a typical creation differential equation.
Here we have production rates dependent on the concentration
of other proteins. The processes related to consumption are not totally
understood (see Martinez-Vincente et al). We understand cell growth, as
distinct from mitotic duplication, but the growth of a cell is merely the
expansion of what was already in the cell when at the end of its mitotic
creation. In contrast, we understand apoptosis, the total destruction of the
cell, we also understand that certain proteins flow outside the cell or may be
used as cell surface receptors, but the consumption of these is not fully
understood. Yet we can postulate a similar destruction differential equation.
This is based upon the work of Martinez-Vincente et al which
states[1]:
All intracellular proteins undergo continuous synthesis
and degradation. This constant protein turnover, among other functions, helps
reduce, to a minimum, the time a particular protein is exposed to the hazardous
cellular environment, and consequently, the probability of being damaged or
altered. At a first sight, this constant renewal of cellular components before
they lose functionality may appear a tremendous waste of cellular resources.
However, it is well justified considering the detrimental
consequences that the accumulation of damaged intracellular components has on
cell function and survival. Furthermore, protein degradation rather than mere
destruction is indeed a recycling process, as the constituent amino acids of
the degraded protein are reutilized for the synthesis of new proteins.
The rates at which different proteins are synthesized and
degraded inside cells are different and can change in response to different
stimuli or under different conditions. This balance between protein synthesis
and degradation also allows cells to rapidly modify intracellular levels of
proteins to adapt to changes in the extracellular environment. Proper protein
degradation is also essential for cell survival under conditions resulting in
extensive cellular damage. In fact, activation of the intracellular proteolytic
systems occurs frequently as part of the cellular response to stress. In this
role as ‘quality control’ systems, the proteolytic systems are assisted by
molecular chaperones, which ultimately determine the fate of the
damaged/unfolded protein.
Damaged proteins are first recognized by molecular
chaperones, which facilitate protein refolding/repairing. If the damage is too
extensive, or under conditions unfavorable for protein repair, damaged proteins
are targeted for degradation. Protein degradation is also essential during
major cellular remodeling (i.e. embryogenesis, morphogenesis, cell
differentiation), and as a defensive mechanism against harmful agents.
We have also discussed this process with regards to the
function of ubiquitin, which marks proteins for elimination. As Goldberg states[2]:
Proteins within cells are continually being degraded to
amino acids and replaced by newly synthesized proteins. This process is highly
selective and precisely regulated1,
and individual proteins are destroyed at widely different rates, with
half-lives ranging from several minutes to many days. In eukaryotic cells, most
proteins destined for degradation are labelled first by ubiquitin in an energy requiring
process and then digested to small peptides by the large proteolytic complex,
the 26S proteasome.
Indicative of the complexity and importance of this
system is the large number of gene products (perhaps a thousand) that function
in the degradation of different proteins in mammalian cells. In the past
decade, there has been an explosion of interest in the ubiquitin–proteasome
pathway, due largely to the general recognition of its importance in the
regulation of cell division, gene expression and other key processes1. However, the cell’s degradative machinery must
have evolved initially to serve a more fundamental homeostatic function — to
serve as a quality-control system that rapidly eliminates misfolded or damaged
proteins whose accumulation would interfere with normal cell function and
viability.
Also we refer to the recent review work of Ciechanover which
details the evolution of this understanding[3].
In contrast the proteins are consumed and thus the negative
sign. In toto we have a combined equation as a total balance of proteins. This
assumes we have a production mechanism for each of the proteins, namely their
genes and the activators and repressors as required.
d. Pathway Dynamics must be meaningful. Let us consider the
pathway as shown below. This is a typical melanoma pathway we have shown
before.
Now let us consider PTEN blocking BRAF and Akt. Now
physically it is one molecule of PTEN needed for each molecule of BRAF and
PI3K. But what if we have the following: n(PTEN)n(PI3K).
Here we have PTEN blocking some but not all the BRAF and
PTEN blocking all the PI3K. At least at time t. Do we have an internal
mechanism which then produces even more PTEN? One must see here that we are
looking at the actual numbers of PTEN, real numbers reflecting the production
and destruction rates. We know for example that if we have a mutated BRAF then
no matter how much PTEN we have an unregulated pathway.
Now it is also important to note that this “model” and
approach is distinct in ways from classic kinetics, since the classic model
assume a large volume and concentrations in determining kinetic reaction rates
of catalytic processes. Here we assume a protein binds one on one with another
protein to facilitate a pathway.
Thus knowing the dynamics of individual proteins, and
knowing the pathways of the proteins, namely the temporary adhesion of a
protein, we can determine several factors:
1.
The number of free proteins
by type
2.
The pathways activated or
blocked
3.
The resultant cellular
dynamics based on activated pathways.
It should be noted that we see pathways being turned on and
off as we produce and destroy proteins. There is a dynamic process ongoing and
it all depends on what would be a stasis level of proteins by type. The
question is; are cells in stasis or are they in a continual mode of regaining a
temporary stasis?
This also begs the question, that if as we have argued, that
cancer is a loss of stasis due to pathway malfunction, then can this be a
process of instability in the course of a normal cell? Namely is there in the
dynamics of cell protein counts, unstable oscillator type modes resulting in
uncontrolled mitotic behavior. Namely can a cell get locked into an unstable
state and start reproducing itself in that state, namely an otherwise normal
cell.
e. Total intracellular dynamics can be modeled yet the
underlying processes are still not understood and the required measurements are
yet to be determined.
Here we look at the intercellular dynamics as well, not just
as a stand-alone model. By this methodology we look at intercellular
communications by ligand binding and the resulting activation of the
intracellular pathways. We must consider both the intercellular signalling
between like cells but also between unlike, such a white cells perhaps as
growth factor inhibitors and the like. We also then must consider the
spatiodynamics, namely the “movement” of the cells, or in effect the lack of
fixedness or specificity of function. This becomes a quite complex problem.
There are two functions we examine here:
a. Intercellular binding or adhesion: E cadherin is one
example that we see in melanocytes. Pathway breakdown may result in the
malfunctioning of E cadherin.
The above demonstrated E cadherin in melanocyte-keratinocyte
localization. The bonds are strong and this stabilizes the melanocyte in the
basal layer. If however the E cadherin is compromised then the bond is broken,
or materially weakened, and the melanocyte starts to wander. Movement for
example above the bottom of the basal layer and upwards is pathognomonic of
melanoma in situ. Wandering downward to the dermis becomes a melanoma. Thus the
pathways activating E cadherin production is one pathway essential in the
inter-cellular dynamics.
b. Ligand production and receptor production: Here we have
cells producing ligands, proteins which venture out of the cell and become
signalling elements in the intercellular world. We have the receptor production
as well, where we have on the surface of cells, various receptors, also
composed of cell generated proteins, which allow for binding sites of the
ligands and result in pathway activation of some type. For example various
Growth Factors, GF proteins, find their way to receptors, which in turn
activate the pathways. Wnt is an example of one of these ligands which we have
shown above.
It can also be argued that as ligands are produced and as
the “flow” throughout the intercellular matrix, we can obtain effects similar
to those in the Turing tessellation models. Namely a single ligand may be
present everywhere but density of ligands may vary in a somewhat complex but
determinable manner, namely is a wavelike fashion.
This is akin to the Turing model used in patterning of
plants and animals[4].
Namely the concentration of a ligand, and in turn its effect, may be controlled
by
In this case we would want a model which reflects the total
body spatiotemporal dynamics This type of models is an ideal which may or may
not be achievable. In a simple sense it is akin to diffusion dynamics, viewing
the cancer cells as one type of particle and the remaining body cells as
another type. The cancer cells have intercellular characteristics specific to
cancer and the body cells have functionally specific characteristics. Thus we
could ask questions regarding the “diffusion” of cancer cells from a local
point to distant points based upon the media in between. The “rate” of such
diffusion could be dependent upon the local cells and their ability for example
to nourish the cancer cells as well. In this model we could define an average
concentration of cancer cells at some position x and time t and we would have
some dynamic process as well.
This is a diffusion like equation and is a whole body
equation. Perhaps knowing what the rate of diffusion is on a cell by cell basis
may allow one to determine the most likely diffusion path for the malignancy,
and in turn direct treatment as well.
This is of course pure speculation since there has been to
my knowledge any study in this area. Except one could imagine a system akin to
PET scans and the like which would use as input the surface markers from a
malignancy and then the body diffusion rates to plot out in space and time the
most likely flow of malignant cells and thus plan out treatment strategies.
Although this model is speculative we shall return again to it in a final
review of such models since it does present a powerful alternative.
This concept of total cellular dynamics is in
contradistinction to the intercellular transport. In the total cellular
dynamics model we regard the model as one considering the flow of altered cells
across an existing body of stable differentiated cells.
We may then ask, what factors drive cancer cells to what
locations? One may putatively state that cancer cells will follow the path of
least resistance and/or will proceed along “flow lines” consistent with what
propagation dynamics they may be influenced by.
The concept of a model of Total Cellular Dynamics is
somewhat innovative. It focuses on the movement of the cancer cells throughout
the body. We will consider three possible possibilities:
1. No Stem Cells
2. Stem Cells but Fixed at Initial Location
3. Stem Cells which are mobile.
In Case 1 all malignant cells are clones of each other at
least at the start. As the malignant cells continue through mitosis additional
mutations are likely so that after a broad set of mitotic divisions we have a
somewhat heterogeneous set of malignant cells, some more aggressive than
others. As with most such cancer cells they also produce ligand growth factors
which stimulate each other and result in the cascade of unlimited growth and
duplication.
In Case 2 we assume that there was a single cell which
mutated and that this becomes the CSC. The CSC replicates producing one CSC for
self-replication and TICs which migrate. We assume that the CSC may from time
to time actually double, but not at the mitosis rate of the base. Furthermore
we assume the CSC sends out growth factors, GF, to the TICs. The GF flow
outward in a wave like manner from the somewhat position stabilized CSCs to the
TICs which are mobile and both diffuse and flow throughout the body. The GF
must find the TICs which become a distant metastasis.
In Case 3 in contrast to Case 2, we assume mobile CSC and
thus the CSCs also flow according to some set of rules.
Now depending on the case we assume we can model the flow of
cancer cells according to some simple dynamic distributed models[5]. Thus we could have[6] a partial differential
equation of the type found in McGarty (see White Paper).
This provides diffusion, flow, and rate elements. The rate
term, the F term, is a rate of change in time at a certain location and time
specific. It is the duplication rate at that specific location due to the
normal mitotic change. The last term may be both pathway and environment
driven.
Now this description has certain physical realities.
Here above we describe the three factors in terms of their
effects and their causes. The three elements of the equation; diffusion, flow,
and growth, are the three ways in which cancer cells move. We can summarize
these as below:
Factor
|
Diffusion
|
Flow
|
Growth
|
Physical Effect
|
Cancer cells begin to diffuse
due to concentration effects.
|
Cancer cells are “forced” to
move by a flow mechanism driven them in a direction along flow lines.
|
Cancer cells begin to go through
mitosis and cell growth.
|
Genetic Driver
|
Movement is due to the loss of location
restrictors such as E cadherin found in melanocytes and restricting their
movement.
|
Flow lines may be developed by means of
metabolic needs of the cell in search of the nutrients required for growth.
This may be a combination of angiogenesis as well as a Warburg like effect.
|
Growth factor ligands attach to the surface
of the cell. Flow of such ligands and their production may be influenced by a
Turing flow effect thus accounting for complexity of location of growth.
|
Impact
|
Slow migration in local areas.
|
Cells have lost functionality
and move to maximize their nutrition input to facilitate growth.
|
Cancer cells may find optimal
areas for proliferation based upon factor related to ligand density.
|
Now consider the following graphic as a human body,
We have a D, E, F, for each gross portion of the body. We
also have a model as specifically below in the Table:
Organ
|
D
Diffusion
|
E
Flow
|
F
Production
|
Epidermis
|
0.5
|
0.01
|
0.7
|
Dermis
|
0.4
|
0.02
|
0.5
|
Cutis
|
0.3
|
0.05
|
0.2
|
Blood
|
5.0
|
0.5
|
0.01
|
Brain
|
0.1
|
0.01
|
0.2
|
Liver
|
2.0
|
0.2
|
0.3
|
Lung
|
3.0
|
0.3
|
0.4
|
Kidney
|
1.5
|
0.4
|
0.5
|
Bone
|
2.5
|
0.5
|
1.0
|
The above numbers are purely speculative. But if we can
ascertain them then we get a solution of p(x,t) in time. Note that here we have
a two dimensional space. Thus we have the above constants applying only to this
artifactually spatial model. Distance is measured in terms of movement across
the interfaces. For simplicity we assume that all other space is impenetrable
by any means. This we have production, flow and diffusion in each area.
Note that in the above we have laid out the x and y
coordinates such that we have blood flow in the center, namely the metastasis
flows via blood, and then enters organs as shown. The “location” of the organs
are distances. Note also the origin of the malignancy is at (0,0).
Now we can relate the constants to the pathway distortions
which are part of the malignancy as well.
The question is how do we determine these constants so that
we may verify the model. Let us assume we can do so via examination of prior
malignancy, not an obvious task but one we shall demonstrate. One must be
cautious also to include in the determination pathway factors for each
malignancy and its state and stage. Thus the three constants will be highly
dependent upon the specific genetic makeup of the initial malignancy.
Turing Tessellation
In 1952 Alan Turing, in the last year and a half of his
life, was focusing on biological models and moving away from his seminal
efforts in encryption and computers. It was Turing who in the Second World War
managed to break many of the German codes on Ultra and who also created the
paradigm for computers which we use today. In his last efforts before his
untimely suicide Turing looked at the problem of patterning in plants and
animals. This was done at the same time Watson and Crick were working on the
gene and DNA. Turing had no detailed model to work with, he had no gene, and he
had just a gestalt, if you will, to model this issue. Today we have the details
of the model to fill in the gaps in the Turing model.
The Turing model was quite simple. It stated that there was
some chemical, and a concentration of that chemical, call it C, which was the
determinant of a color. Consider the case of a zebra and its hair. If C were
above a certain level the hair was black and if below that level the hair was
white. As Turing states in the abstract
of the paper:
"It is suggested that a system of chemical
substances, called morphogens, reacting together and diffusing through a
tissue, is adequate to account for the main phenomena of morphogenesis. Such a
system, although it may originally be quite homogeneous, may later develop a
pattern or structure due to an instability of the homogeneous equilibrium,
which is triggered off by random disturbances. Such reaction-diffusion systems
are considered in some detail in the case of an isolated ring of cells, a
mathematically convenient, though biologically unusual system.
The investigation is chiefly concerned with the onset of
instability. It is found that there are six essentially different forms which
this may take. In the most interesting form stationary waves appear on the
ring. It is suggested that this might account, for instance, for the tentacle
patterns on Hydra and for whorled
leaves. A system of reactions and diffusion on a sphere is also considered. Such
a system appears to account for gastrulation. Another reaction system in two dimensions
gives rise to patterns reminiscent of dappling. It is also suggested that
stationary waves in two dimensions could account for the phenomena of
phyllotaxis.
The purpose of this paper is to discuss a possible
mechanism by which the genes of a zygote may determine the anatomical structure
of the resulting organism. The theory does not make any new hypotheses; it
merely suggests that certain well-known physical laws are sufficient to account
for many of the facts. The full understanding of the paper requires a good
knowledge of mathematics, some biology, and some elementary chemistry. Since
readers cannot be expected to be experts in all of these subjects, a number of
elementary facts are explained, which can be found in text-books, but whose
omission would make the paper difficult reading."
Now, Turing reasoned that this chemical, what he called the
morphogen, could be generated and could flow out to other cells and in from
other cells. Thus focusing on one cell he could create a model across space and
time to lay out the concentration of this chemical. He simply postulated that
the rate of change of this chemical in time was equal to two factors; first the
use of the chemical in the cell, such as a catalyst in a reaction or even part
of the reaction, and second, the flow in or out of the cell. The following
equation is a statement of Turing's observation.
It allows one to solve for a concentration, C, as a function of time
and space. It requires two things. First is the diffusion coefficient to and
from cells and second the functional relationship which shows how the chemical
is used within a cell.
The question now is how does one link the coefficients in
the models. For example if we believe that diffusion D depends on E cadherin
concentration, namely as E cadherin decreases then D increases we may postulate
a simple linear relationship between diffusion constants and protein
concentrations, where the constants are to be determined. We know that the more
E cadherin the stickier is the cell and the less diffusion that occurs. Thus
the above is at the least a first order approximation. In a similar manner we
can relate F to PTEN and p53.
This is merely suppositional. But we do know the following:
1. The genes which are expressed for adhesion and
replication are known.
2. We know the pathways for these genes
3. We know the intracellular models controlling these genes.
4. We know that functionally an excess or paucity of a gene
has a certain effect.
5. We know that in general in small amounts the world is
linear.
6. We know that we can use regression techniques based upon
collected data to determine coefficients in a general sense.
Thus we have a fundamental basis to express the relationships
for all gross constants in terms of linearized versions of the protein
concentrations.
Now we have related intracellular concentrations, which
themselves may be temporally and spatially dependent, to the total parameter
values for the flow of cells throughout the body. We may also want to relate
these to organ specific parameters as well.
Thus what we have achieved is as follows:
1. Model relating intracellular and whole body.
2. Methodology to determine the constants.
3. Methodology to go from patient data to prognostic data.
4. Methodologies to establish possible treatment
methodologies. Namely what gene controls will result in what whole body
reactions.
We can now summarize this models we have considered. First
we should emphasize that for the most part those working in the field have
developed pathway models which exhibit a non-temporal mode, it is some steady
state model, and the model assumes a protein to protein connection, as if there
were a single protein molecule produced and that the interacting proteins were
there or not. Part of the simplicity of the models is determined by the limits
of what can be measured. We have herein attempted not to limit the results by
what can be accomplished currently but has extended the model to levels which
assist in a fuller representation of reality. However even here we may very be
falling short.
For we have deliberately neglected such things as miRNA,
methylation, and the stem cell paradigm just to name a few.
We combine all four methods in a graphic below. We summarize
the key differences and differentiators. Currently most of the analytical
models focus on pathways. This can generally be supported by means of microarray
technology and even rough estimates of relative concentrations may be inferred
by such an approach.
The risks we see even in the above models is the absence of
exogenous epigenetic factors and the inclusion of a stem cell model. The latter
issue is one of major concern. For example if we have true cancer stem cells,
CSC, then we have a proliferation of differing cell types. The use of
microarrays is for the most part and averaging methodology, not a cell by cell
methodology. If we collect cells from say a melanoma tumor. how much of that is
a CSC and how much a TIC. And frankly should we identify CSCs only and perform
our analysis on those cells alone.
[3]
Ciechanover , A, Intracellular Protein Degradation: From a Vague Idea through
the Lysosome and the Ubiquitin-Proteasome System and onto Human Diseases and
Drug Targeting, Rambam Maimonides Medical Journal, January 2012, Volume 3,
Issue 1
[4] Turing, A., The Chemical Basis of Morphogenesis,
Phil Trans Royal Soc London B337 pp 37-72, 19459.
[5] See Andersen p 277 of Bellomo et al for an variant on
what we are proposing here. The Andersen model is somewhat similar but lacks
the detail we present herein. Also there is in the same volume a paper by
Pepper and Lolas focusing on the dynamics of the lymphatic cancer system, p
255. Bellomo, N., et al, Selected Topics
in Cancer Modeling, Birkhauser (Boston) 2008.
[6] McGarty, T., Stochastic Systems and State Estimation, Wiley
(New York) 1974.
1. Szallasi, Z. System Modeling in Cellular Biology: From
Concepts to Nuts and Bolts. MIT Press (Cambridge) 2006.
1. Klipp, E., et al, Systems
Biology, Wiley (Weinheim, Germany) 2009.
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