Solving the Puzzle of Metastasis: The Evolution of Cell
Migration in Neoplasms
Jun Chen1, Kathleen Sprouffske1,2, Qihong Huang1,2,3, Carlo C. Maley4*
1Genomics and Computational Biology Program, School of Medicine, University of Pennsylvania, Philadelphia, Pennsylvania, United States of America, 2Molecular and
Cellular Oncogenesis, The Wistar Institute, Philadelphia, Pennsylvania, United States of America, 3Cell and Molecular Biology Program, School of Medicine, University of
Pennsylvania, Philadelphia, Pennsylvania, United States of America, 4Helen Diller Family Comprehensive Cancer Center and Department of Surgery, University of
California San Francisco, San Francisco, California, United States of America
Background: Metastasis represents one of the most clinically important transitions in neoplastic progression. The evolution
of metastasis is a puzzle because a metastatic clone is at a disadvantage in competition for space and resources with non-
metastatic clones in the primary tumor. Metastatic clones waste some of their reproductive potential on emigrating cells
with little chance of establishing metastases. We suggest that resource heterogeneity within primary tumors selects for cell
migration, and that cell emigration is a by-product of that selection.
Methods and Findings: We developed an agent-based model to simulate the evolution of neoplastic cell migration. We
simulated the essential dynamics of neoangiogenesis and blood vessel occlusion that lead to resource heterogeneity in
neoplasms. We observed the probability and speed of cell migration that evolves with changes in parameters that control
the degree of spatial and temporal resource heterogeneity. Across a broad range of realistic parameter values, increasing
degrees of spatial and temporal heterogeneity select for the evolution of increased cell migration and emigration.
Conclusions: We showed that variability in resources within a neoplasm (e.g. oxygen and nutrients provided by
angiogenesis) is sufficient to select for cells with high motility. These cells are also more likely to emigrate from the tumor,
which is the first step in metastasis and the key to the puzzle of metastasis. Thus, we have identified a novel potential
solution to the puzzle of metastasis.
Citation: Chen J, Sprouffske K, Huang Q, Maley CC (2011) Solving the Puzzle of Metastasis: The Evolution of Cell Migration in Neoplasms. PLoS ONE 6(4): e17933.
Editor: Marc Tjwa, University of Frankfurt - University Hospital Frankfurt, Germany
Received September 4, 2010; Accepted February 17, 2011; Published April 27, 2011
Copyright: ? 2011 Chen et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits
unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Funding: This work was supported in part by National Institutes of Health (http://nih.gov/) (R01 CA119224, R03 CA137811, P01 CA91955, R01 CA140657 to CCM,
R21 NS059478 to QH, P30 CA010815 to CCM and QH, and T32 HG000046 for KS); Landon AACR Innovator Award for Cancer Prevention (http://www.aacr.org/
home/scientists/research-funding-fellowships/landon-aacr-innovator-award-for-cancer-prevention-research.aspx), Research Scholar Grant #117209-RSG-09-163-
01-CNE from the American Cancer Society (http://www.cancer.org/), the PhRMA Foundation (http://www.phrmafoundation.org/) and a Mclean contributionship
(http://foundationcenter.org/grantmaker/mclean/) for CCM; National Institutes of Health (http://nih.gov/) (R01CA148759), the Breast Cancer Alliance (http://
www.breastcanceralliance.org/), Edward Mallinckrodt Jr. Foundation (http://fundingopps.cos.com/cgi-bin/getRec?id=3691) and W. W. Smith (http://www.
wwsmithcharitabletrust.org/) for QH. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
Competing Interests: The authors have declared that no competing interests exist.
* E-mail: firstname.lastname@example.org
Clinically, the evolution of metastasis is one of the most
important transitions in neoplastic progression. Prior to metastasis,
most neoplasms can be cured surgically, and 5-year survival rates
are often above 90%. However, once a neoplasm has spread to
distant sites, some form of systemic therapy is necessary, and 5-
year survival rates often fall below 15% . Understanding and
preventing metastasis would have a dramatic impact on the
management and burden of the disease.
Bernards and Weinberg focused attention on a paradox in our
understanding of the evolution of metastasis . Within a
neoplasm, cells compete for space and resources. (Epi)genetic
instability generates new mutant clones, and those with a survival
or reproductive advantage tend to spread within the neoplasm .
If a cell acquires a mutation that increases the chances that its
offspring will emigrate from the neoplasm, that clone should be at
a disadvantage within the primary neoplasm, because some of its
reproductive potential is lost to emigration . Clones that do not
emigrate will have a net growth advantage over the emigrating
clone, which should be quickly driven extinct [4,5]. However,
evidence suggests that 106–107cells emigrate from a neoplasm
every day yet rarely establish a growing metastasis in a new
location in the body . Thus the evolution of cell emigration
from the primary neoplasm does not seem to be a rate limiting step
in metastasis. How could a metastatic clone ever grow large
enough to produce the millions of emigrating cells necessary to
overcome the low probability of establishing a metastasis?
Four possible, non-mutually exclusive, solutions for the puzzle
of metastasis have been proposed previously. First, a mutation that
provides the potential to metastasize might have other effects
(pleiotropy) that increases the survival or reproductive potential of
the clone and so compensates for the fitness penalty of cell
emigration [2,5]. In a theoretical exploration of the first solution,
Dingli and colleagues  suggested a second solution: there may
be so many cells in a neoplasm that millions of de novo metastatic
PLoS ONE | www.plosone.org1April 2011 | Volume 6 | Issue 4 | e17933
mutants may be produced every cell generation. Even if each
metastatic clone is at a competitive disadvantage and tends to go
extinct, new metastatic clones may continually replace them.
Third, the potential to metastasize might only be triggered late in
progression, by a change in the tumor microenvironment ,
allowing the clone to expand, without the fitness penalty of
emigration, before the change in the microenvironment. Fourth,
an early mutation might confer the potential to metastasize, but
that potential may only be activated by a later mutation .
However, this is not actually a solution because the later mutation
leads to a fitness disadvantage for the metastatic clone and that
clone with both mutations should not expand, which mirrors the
original framing of the problem.
Recently, we identified a fifth alternative based on dispersal
theory in ecology , the ‘‘resource heterogeneity’’ solution.
Dispersal theory predicts that resource heterogeneity in both space
and time selects for migration in organisms  because organisms
that move to locate regions with more resources than their current
location will leave more offspring than sedentary organisms. We
apply dispersal theory  to cancer to solve the paradox of the
evolution of metastasis. There is microenvironmental variability in
neoplasms - regions within a neoplasm can become transiently
hypoxic [9–13] due to poorly regulated angiogenesis, changes in
the vascular architecture and temporary occlusion or interruption
of blood flow by neoplastic cells [13–15]. Thus, we propose that
resource heterogeneity within neoplasms selects for cell ‘‘migra-
tion’’ - or motility - within the neoplasm, and that cell emigration
from the neoplasm - or invasion - is a by-product of that selection.
The puzzle of metastasis was criticized for not being framed
quantitatively . Here we show that a quantitative model can
illustrate a solution to the paradox of the evolution of cell
emigration. Our computational model extends previous models
[4,5,17] by including spatial effects, the dynamics of resources in
that space and the evolution of the migratory phenotype. We
observe the evolution of cell migration and emigration in a
neoplasm under different degrees of temporal and spatial
heterogeneity of resources.
We implemented an agent-based model of a neoplasm
(NetLogo 4.0.2 , source code available upon request from
the corresponding author). The neoplasm is represented as a grid
of patches that store nutrients delivered by blood vessels, while
cells are represented as motile agents that consume nutrients
stored in the patches. For simplicity, the resources in the model are
described as oxygen, though they could alternatively represent
glucose or any other diffusible factor delivered through the
Time is divided into short intervals of 12 hours. During each
time step, blood vessels can form, be occluded by neoplastic cells
proliferating in the confined space , and produce nutrients,
which then diffuse. Cells can consume resources, move, reproduce,
and die. See Figure S1 for a flow chart of a time step and Table S1
for model parameters and their normalized values.
C cells and V blood vessels are positioned in continuous 2D
space atop a square grid of P patches. For each time step, every
patch containing a blood vessel receives riunits of oxygen. To
approximate continuous oxygen dynamics, we performed oxygen
updates tatimes in a time step (Eq. 1). Resource concentrations
at patch j at time t+1 can be described by the following
where there are taresource updates per time step in the model, dc
is the resource diffusion constant, N(j) are the eight adjacent
neighbour patches of j, rais the cell absorption rate, nt
number of cells at position j at time t, riis the resource production
rate for a microvessel and dt
jtakes value 1 if there is a microvessel
at position j at time t and 0 elsewhere. Resource concentrations
cannot become negative because cells are prevented from
absorbing more resources than are present in the location.
Our results are robust with respect to the granularity of the
diffusion dynamics (ta=100, Figure S2). In each update, ri/taunits
of oxygen immediately diffuse throughout the grid by using
NetLogo’s discrete space diffusion function, where each patch
distributes a fraction (dc) of its oxygen to its eight neighboring
patches each iteration. Each cell ciconsumes and stores as much
oxygen as is available to it from its host patch pj, up to ra/taunits.
Next, each cell uses rm/taunits of its stored oxygen nt,ifor its
metabolism, leaving it with nt,i- rm/taunits. When we changed the
number of blood vessels, we adjusted riso that the total input of
oxygen to the system remained constant. The parameters of
oxygen dynamics were set so as to achieve realistic oxygen
gradients around microvessels in normal tissues  (see Table
Any blood vessel vi with more than to cells in its patch is
occluded and removed from the simulation. If the total number of
blood vessels v is less than the equilibrium number V, then V - v
new vessels are added randomly to hypoxic patches with less than
thunits of oxygen and at least one cell, which is required to signal
for angiogenesis. Angiogenesis is a complex process, including the
sensing of hypoxic conditions, release of angiogenic and anti-
angiogenic factors as well as endothelial cell response. The result of
these processes, to a first approximation, is that new blood vessels
grow into areas of hypoxic cells. Because angiogenesis is not the
focus of our model, we have abstracted away most of the
complexities of the process and simply maintain a homeostatic
density of blood vessels, growing new blood vessels in locations
where there are hypoxic cells that would release angiogenic
Next, each cell cimay die (if nt,i=0), reproduce (if nt,i.nr), or
move (if 0,nt,i#nr,). When a cell divides it splits its stored resources
equally between its two daughter cells. During division, in both
daughter cells, the migration propensity pi and maximum
migration distance miare mutated with probability m by drawing
a random number from a truncated normal distribution with
mean pior miand standard deviation sdpor sdm, respectively. A cell
moves with probability pi. Migrating cells move up to mipatches.
These misteps can be taken either randomly (‘‘random migration’’)
or by ascending the local oxygen gradient (‘‘gradient ascent’’) as
both strategies have been observed in neoplasms ; in both
cases, the cell can move to any one of its nine closest patches (its
current patch and eight neighboring patches). If during its
movement it reaches a patch on the edge of the neoplasm, the
cell is removed from the population and recorded as an emigrating
The ‘‘variable lifespan blood vessel’’ model as described results
in spatial and temporal resource heterogeneity because the blood
vessel lifespan varies as a result of local cell dynamics. To tease
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apart the role of spatial and temporal effects in the variable
lifespan blood vessel model, we developed a ‘‘fixed lifespan blood
vessel’’ model in which we controlled more precisely spatial and
temporal dynamics. In this model, blood vessels are destroyed after
a fixed number of time steps tfrather than being occluded by cell
crowding. While the fixed lifespan blood vessel model is not
biologically realistic, it allows us to interpret the results from the
more realistic variable lifespan blood vessel model. ‘‘Static blood
vessels’’ can be simulated by setting the blood vessel lifespan to be
infinity, and ‘‘uniform oxygen input’’ can be simulated by creating
exactly one, static blood vessel per patch. Using this simplified
model, we can control the degree of spatial heterogeneity by
controlling the number of blood vessels, and the degree of
temporal heterogeneity by varying the lifespan of a blood vessel (or
the density of cells that cause occlusion toin the variable lifespan
blood vessel model).
Because the cells evolve to exploit the available resources, there
is not a precise mapping of parameter values to available resource
heterogeneity in the model. We predicted that it is the amount of
available resources that is relevant to the evolution of migration.
Since cells are quickly selected to utilize all available resources,
only the combination of spatial and temporal heterogeneity of
blood vessels in the neoplasm produces transient unutilized
resources. To test this, we measured the average amount of
available resource per patch, over the last 200 time steps of each
model run, and evaluated its relationship to the evolution of cell
We ran the model for 5,000 time steps, ,7 simulated years, to
approximate the time required to develop metastasis [21,22].
Every parameter configuration was replicated 10 times. Data were
collected and averaged over the final 200 time steps of each run. In
each case, we measured the: (1) Mean migration propensity of all
the cells (pi), (2) Mean maximum migration distance (mi) of all the
cells in the neoplasm, and (3) Mean number of cells leaving the
edge of the neoplasm (emigrating cells) per time step. We also
computed the product of the first two parameters, and refer to it as
the ‘‘expected migration distance’’ of the neoplasm, which reflects
the expected distance a cell will travel in one time step. If not
specified explicitly, all following experiments were done under the
random migration strategy.
A t-test was used to test the difference of the mean equilibrium
values between two simulation conditions. Linear regressions were
used to quantify associations between experimentally manipulated
variables (blood vessel number and lifespan) or their outcomes
(observed degree of resource heterogeneity) with the expected
migration distance and number of migratory cells.
Each of the model variants and parameter conditions
corresponds to angiogenesis and cell movement behaviors within
a neoplasm. For each of the experiments that follow, we observe
the evolution of cell migration and emigration from a neoplasm
across the range of parameters.
Because the amount of cell movement within a neoplasm early
in progression is unknown, we tested several reasonable initial
conditions within which a migratory cell could evolve: (1) Cells are
generally stable and do not move, (2) Cells have a low level of
movement within a neoplasm, and (3) Each cell within a neoplasm
can have a different level of motility. We tested these three initial
conditions by initializing (1) All cells with a migration propensity of
0 and a maximum migration distance of 0; (2) All cells with a
migration propensity of 0.05 and a maximum migration distance
of 1; and (3) Cells with random migration propensity values and
maximum migration distance values with uniform probability over
the intervals [0, 0.6] and [0, 6] respectively. We found that the
initial amount of cell motility had no influence on the evolution of
the final levels for both phenotypes (Figure S3; t-test P.0.05 for all
We then compared the evolution of migration under the three
oxygen input methods: ‘‘uniform resource input’’, ‘‘static blood
vessels’’, and ‘‘variable lifespan blood vessels.’’ In the uniform
resource input simulations, each patch received a constant amount
of oxygen input every time step, which resembles the resource
input in normal, adequately oxygenated tissue. In the static and
variable lifespan blood vessel simulations, resources were distrib-
uted via a constant number of blood vessels, which either stayed in
the same locations for the entire run or changed their locations.
The variable lifespan blood vessel model simulates the blood vessel
dynamics observed in a neoplasm. Comparison of the static blood
vessel model to the variable lifespan blood vessel model allows us
to test the effects of temporal heterogeneity on the evolution of cell
motility. A variable lifespan blood vessel was occluded when more
than 20 cells occupied its patch and was replaced by a new vessel
in a hypoxic patch. The value of this parameter did not affect the
qualitative results. Neoplasms with variable lifespan blood vessels
evolved higher values for the migration propensity and maximum
migration distance than the other two resource input models
(Figure 1). The combination of spatial and temporal heterogeneity
generates transient regions of unexploited resources (Figure 2).
Note that static blood vessels do not produce heterogeneity of
unutilized resources because cells proliferate around the blood
vessels until they consume all available resources (Figure 2B).
Uniform input of resources across the entire environment can lead
to more available resources and greater evolution of cell migration
than static blood vessels because the uniform input of resources
leads to fewer cells at each source of resources and so more
stochasticity of cell dynamics in each patch. The spatial
heterogeneity generated by blood vessels leads to patches of
necrosis and hypoxia that are typical of a neoplasm (Figure 2D).
We ran the fixed lifespan blood vessel model with number of
blood vessels from 30 to 600 and vessel lifespans from 4 to 500
time steps on a log scale, which covered the ranges observed in real
neoplasms (Table S1). Decreasing blood vessel numbers and
lifespan (increasing spatial and temporal heterogeneity) selected for
increased cell migration and number of emigrating cells (Figure 3;
linear regressions P,0.001). The emigrating cell number and
expected migration distance are closely correlated (r =0.984,
We repeated the experiments under the condition in which cells
climb up resource gradients  (Figure 4). If a cell reached a local
maximum of resources, it remained there even if it had the
capacity to move further. Here, resource heterogeneity still selects
for cell migration with the similar pattern as in neoplasms using
random migration strategy (Figure 4). The association between
resource heterogeneity (both temporal and spatial) and the
expected migration distance remains strong (linear regression
P,0.001), as does their relationship with the number of cells that
migrate off the edge of the neoplasm (linear regression P,0.001).
In a real neoplasm, resource concentrations may increase at the
borders of the neoplasm (and other routes of exit) and so migratory
cells following those gradients may exit the neoplasm more
frequently than we have represented in our model.
We found that the fewest number and shortest lifespan of blood
vessels led to the maximal amount of available, unutilized
resources, which was statistically significantly associated with the
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evolution of expected migration distance (Figures 5A and 5B;
r=0.79 for random migration, r=0.81 for gradient ascent,
P,0.001 for both), and the number of emigrating cells
(Figures 5C and 5D; r=0.74 for random migration, r=0.79 for
gradient ascent, P,0.001 for both). Figure S4 shows snapshots of
available resources at the four extreme settings of the number of
blood vessels and their lifespans.
In the model, the parameter values for the mutation standard
deviation of migration propensity (sdp) and maximum migration
distance (sdm) affect the rate of evolution and are set empirically.
To see whether sdpor sdmaffect the above results, we ran the model
with different values of sdp(0.01, 0.1) and sdm(0.1, 1). Though the
final evolved values for these migration phenotypes vary with
different sdpor sdm, neoplasms with variable lifespan blood vessels
still evolved higher values for all migration phenotypes (Figure S5).
To explore the interaction of selection for migration with
selection for proliferation, we allowed cells to evolve the ability to
divide with fewer resources. Increased proliferation is a hallmark
of carcinogenesis . When cells could optimize their prolifer-
ation under restricted resources, cell migration rates evolved to
even higher levels than before. Neoplasms with variable lifespan
blood vessels quickly evolved a lower threshold of resources
necessary to divide (increased their proliferation rate), compared to
uniform resource input or static blood vessels (Figure S6).
All the prior experiments were run on a grid of 4,096 patches.
We tested neoplasms of 1024, 2116, 4096, 8100, and 16,384
patches to determine how the dynamics scale with the size of the
simulated neoplasm. The fixed lifespan blood vessel model, with 6
time step lifespans, was used in these simulations. Blood vessel
numbers were scaled with neoplasm size from 25, 50, 100, 200, up
Figure 1. Examples of evolution of migration with dynamic blood vessels, uniform input, and static blood vessels. (A) Evolution of the
migration propensity. (B) Evolution of the maximum migration distance per time step. (C) Evolution of the expected migration distance (the product
of migration propensity and maximum migration distance). (D) The number of migratory cells leaving the neoplasm per time step. In the static blood
vessel model (dotted lines), 100 blood vessels remain fixed in position throughout the run of the model. In the dynamic model with variable lifespan
blood vessels (solid lines), a blood vessel was occluded when there were more than 20 cells in its location (patch) and replaced by a new blood vessel
in a hypoxic location. In the uniform input model (dashed lines), all patches received an equal amount of resources each time step.
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to 400 respectively, to ensure that the blood vessel density was the
same regardless of neoplasm size. Each blood vessel delivered the
same amount of resources per time step, regardless of the size of the
neoplasm, so that larger neoplasms received more total resources
and could support a larger population of neoplastic cells. Resource
heterogeneityand expected migration distance appear tobesensitive
to boundary effects in small simulations, but approach an
equilibrium value (and have lower variance) in simulations of
.8,000 patches (Figures S7A and S7B). The emigrating cell number
scales linearly with neoplasm size, and so the frequency of cell
emigration is stable over changes in neoplasm size (Figures S7A and
S7C). Selection for cell migration by temporal and spatial resource
heterogeneity holds regardless of the size of the simulated neoplasm.
Using a computational model, we explored a possible solution to
the paradox of the evolution of metastasis identified by Bernards
and Weinberg. We propose that resource heterogeneity selects
for cell motility, which leads to emigration from the primary
tumor. Our model captures the fitness disadvantage associated
with cell migration in that emigrating cells are removed from the
model. Intriguingly, we observed that these same ‘‘disadvanta-
geous’’ clones were favored in conditions predicted by dispersal
theory in ecology. Namely, we have shown that spatial and
temporal resource heterogeneity selects for cell migration within a
neoplasm, and as a by-product, emigration from a neoplasm.
Figure 2. Resource and cell densities in the model. Green circles show the position of blood vessels and resource density is represented on a
continuum from blue (low) to white (high). When resources flow into the tissue uniformly (A) or through static blood vessels (B), the cells consume all
of the resources and the spatial heterogeneity of unutilized resources is low. When blood vessels are dynamic due to occlusion and angiogenesis (C),
heterogeneity of available resources is greater because there is a lag time between the appearance of a new blood vessel and increased density of
cells in that locale. This explains the differences in the evolution of cell migration for the different resource input modes shown in Figure 1. When
resources flow into a tissue through sparse blood vessels, patches of normoxia and hypoxia lead to corresponding regions of high densities of cells as
well as necrotic regions. Panel D shows an overlay of the cell density for the blood vessels and resources of panel C. The brightness of the cells in
panel D represents the amount of resources each cell has accumulated.
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Specifically, the migration propensity, the maximum migration
distance within a neoplasm, and the resulting number of
emigrating cells, were maximized when there were only a few
blood vessels in the model and when the location of those resource
rich patches changed frequently (Figures 3, 4, 5) as is thought to
occur in neoplasms[9–12,14,15,24]. Oxygen levels can fluctuate in
neoplasms over a period of 109s of minutes, in a spatially
heterogeneous manner, the details of which vary between
neoplasms [10,12]. Transient hypoxia has been observed to occur
over periods of minutes to hours  and chronic hypoxia over
longer time scales . Thus, our simulation results confirm that
selection for migration within a neoplasm under resource
heterogeneity can result in increased levels of cell emigration
from the neoplasm, providing support for the resource heteroge-
neity solution to the paradox of the evolution of metastasis.
The resource heterogeneity solution to the paradox of
metastasis is consistent with a variety of experimental observations,
including spatial and temporal patterns of tumor invasion, patterns
of gene expression in the primary tumor that predict metastasis,
and the metastatic effects of hypoxia on neoplasms. Our model is
consistent with observations of rapid metastasis once a neoplasm
becomes malignant , because we predict that there has been
selection for cell migration prior to invasion. In gene expression
studies, primary neoplasms often exhibit an expression signature of
Figure 3. Relationship between resource heterogeneity and selection for migration when cells move randomly. Spatial heterogeneity
is varied with the number of blood vessels providing resources. Temporal heterogeneity is determined by varying the lifespan of the blood vessels. As
temporal and spatial heterogeneity increased the product of migration propensity and maximum migration distance (expected migration distance)
increased (A) as did the number of cells emigrating from the neoplasm (B). The amount of transiently unutilized, available resources is also maximized
by increasing spatial and temporal heterogeneity (C).
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metastasis [25–29]. Since expression arrays measure the most
common clones in the neoplasm, this has been interpreted as
evidence that a metastatic phenotype often evolves early in
neoplastic progression . These gene expression profiles may
actually be a signature of resource heterogeneity or of migratory
clones. For example, cell motility and stress response genes were
enriched in primary neoplasms associated with recurrence .
Hypoxia has also been associated with increased risk of metastasis
[9,30–33]. The resource heterogeneity hypothesis predicts that
temporal variation in hypoxia should select for increased
emigration, and this is consistent with observations in mouse
models [34–36]. Intriguingly, a molecular mechanism connects
hypoxic stress and migration through HIF1-a , suggesting that
natural selection could co-opt and optimize the (epi)genetics of
cells under hypoxic stress to increase cell migration.
The paradox of the evolution of metastasis depends on the
observation that emigration is a competitive disadvantage for
clones in the primary tumor, and so natural selection should
suppress cell emigration. The following steps in metastasis (e.g.,
survival in the blood, invasion and establishment in a new location,
etc. ) are all selectively advantageous for the emigrating clone,
and so are not paradoxical. We have focused here on the evolution
of the first step of metastasis: migration of cells within the
neoplasm, which leads to emigration from the neoplasm, or
Figure 4. Relationship between resource heterogeneity and selection for migration when cells move up resource gradients. Spatial
heterogeneity was controlled by varying the number of blood vessels and temporal heterogeneity was controlled by varying their lifespan. As with
random movement, increasing temporal and spatial heterogeneity both select for increasing cell migration (expected migration distance); A) and cell
emigration from the neoplasm (B). The combination of both spatial and temporal heterogeneity leads to increased amounts of transiently unutilized,
available resources (C).
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invasion in our model. The paradox of metastasis hinges on this
The possible resolutions to the paradox of metastasis are distinct
in our model, including that metastatic mutations may also
increase fitness, the mutation rate is high enough to generate the
metastatic cells de novo, and microenvironmental changes ‘‘acti-
vate’’ a previously neutral mutation late in progression. In our
model, mutations only affect the propensity or speed of migration,
and do not directly affect apoptosis or proliferation. De novo
migratory mutations cannot explain the evolution of high rates of
migration observed in our models. That being said, the other
solutions to the puzzle of metastasis are not mutually exclusive
with each other or our proposal. There is evidence that some
mutations that facilitate metastasis may also increase the fitness of
the mutant clone [39,40].
Our model is clearly a simplification of intra-tumor dynamics.
In a real neoplasm, cells are likely to emigrate through lymphatic
and blood vessels, not just by leaving the borders of the neoplasm.
Incorporating those details into our model would likely increase
the number of emigrating cells, consistent with the behavior of our
One of the weaknesses of models of metastasis is the lack of
experimental data on cell migration and hypoxia, particularly at
the single cell level in vivo. Thus, we have used a quantitative model
to explore the puzzle of metastasis and develop a hypothesis that
can explain current data and be used to guide future experiments.
Our model supports previous predictions  that assays of spatial
and/or temporal heterogeneity of available resources in a
neoplasm  should predict the risk of metastasis. Spatial
statistics of patchiness could be applied to assays of hypoxia,
Figure 5. Available resources, those not currently being fully exploited by the cell population, select for increased cell migration.
The expected migration distance that evolved (A, B) was closely correlated with the average amount of transiently unutilized, available resources per
patch. Similarly, the number of cells emigrating from the neoplasm was also highly correlated with amount of available resources (C, D). This was true,
regardless of whether cell migration was random (A, C) or by gradient ascent (B, D).
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glucose or other limiting resources in tissue sections [9–13]. We
also predict that direct measures of cell migration in the primary
tumor, perhaps through measures of genes expression and proteins
in cell migration pathways, should be good biomarkers for the risk
There are a number of additional experimentally testable
predictions from our model. First, our model suggests that we
should find greater expression of migration related proteins in
neoplasms with regions of hypoxia compared to neoplasms with
uniform oxygenation. If the half-lives of hypoxia inducible markers
are significantly longer than the rate of cell movement, migratory
cells with those markers might be detected as recent arrivals in
normoxic regions. With a fast enough molecular clock, perhaps
through methylation of CpG sites , one may be able to show
more mixing of cell lineages due to migration in a neoplasm with
resource heterogeneity compared to neoplasms with uniform
resources which should contain contiguous regions of closely
Interestingly, results from our model of resource heterogeneity
suggest a potential strategy for preventing or delaying cancer:
normalizing the resources available to a neoplasm, over space or
time, should tend to reduce the risk of metastasis. In fact, it has
recently been shown that restoration of neoplasm oxygenation
suppresses metastasis . Our model also predicts that cycles of
anti-angiogenic drugs applied to a pre-malignant neoplasm may
select for a metastatic clone and so we should be cautious in the
application of such drugs for cancer prevention [44,45]. It has
recently been shown that a decrease in tumor vascularity is
correlated with tumor invasion in gliobalstoma patients treated
with anti-angiogenesis therapy . These results are consistent
with our model results that migration increases when the vessel
density is decreased. Nevertheless, there is both theoretical and
experimental support for anti-angiogenic therapy in malignant
neoplasms . In fact, constant, low doses of anti-angiogenic
drugs have been shown normalize the vascular networks within
neoplasms [48–50]. Thus, the chronic application of such drugs
may be a route to normalizing the spatial and temporal resources
of a neoplasm, thus preventing selection for cell migration and
In a related model, Bearer et al. studied the effects of resource
heterogeneity and competition between a low- and high-grade
clone on tumor morphology and came to a similar conclusion.
This model represented physical and chemical constraints, along
with cell adhesion dynamics to predict how the interface between
tumor and normal tissue changes over time. In this model, cell
migration was a cellular response to hypoxia and did not evolve. In
contrast, our model does not represent the boundary between
tumor and normal tissue, and instead focuses on the selective
effects of resource heterogeneity on cell migration within the
primary tumor. They found that resource heterogeneity was
amplified by cellular proliferation and migration, leading to
invasive tumor morphologies. From this complementary ap-
proach, they also concluded that normalization of resources
should help suppress invasion. In their case, because resource
homogeneity leads to physical constraints on tumor shape whereas
in our case, resource homogeneity suppresses natural selection for
We have provided a quantitative model for the evolution of cell
migration and emigration from neoplasms that provides a solution
to the puzzle of metastasis. Results from the model are consistent
with both expression signatures of metastasis in primary neoplasms
[25–29] and the observed association between hypoxia and
metastasis [9,30–32]. We propose that cell emigration from a
neoplasm is a side effect of selection for migration within a
neoplasm. The results of our model do not rely upon the exact
details of the model. Regardless of the precise parameters chosen,
the result still holds that resource heterogeneity in space and time
select for cell migration (see Figures S2 through S7). The
predictions of our model are supported by in vivo experiments
[34–36] and clinical results [9,30–32]. We hope that an
understanding of the evolutionary forces that select for metastasis
will be useful for the future prevention of metastasis.
cells and blood vessels for each time step. The labeled
parameters are defined in Table S1. Actions made by cells are
ovals, actions made by vessels are parallelograms, and decision
points are diamonds.
A flow chart outlining the decisions made by
dynamics (input, diffusion, cell uptake, and cell metab-
olism) had no effect on the results that temporal and
spatial heterogeneity select for increased cell migration.
Here, instead of 10 resource dynamic iterations per cell time step,
we used 100 resource iterations (ta). The expected migration
distance (A), emigrating cell number (B) and transiently unutilized,
available resources (C) are all strongly affected by both spatial and
temporal heterogeneity of the resource inputs (blood vessel
number and lifespan). The strong correlation between available
resources and both expected migration distance (D) and the
number of cells that leave the neoplasm (E) remains the same as
Scaling the granularity of the resource
the maximum migration distance phenotypes on the
evolution of migration in example runs of the model. The
random initial phenotypes (blue dotted lines show the population
average) initialize each cell with a migration propensity randomly
chosen from 0 to 0.6 and a maximum migration distance
randomly chosen from 0 to 6, with uniform probability. All cells
in the uniform initial phenotypes case (red lines show the
population average) were initialized with a migration propensity
of 0.05 and maximum migration distance of 1 patch. In all 4
panels, the neoplasm evolves to approximately the same value
indicating initial phenotypes have no impact on the outcome of the
model (t-test across runs of the average values over the last 200
time steps, p.0.05). (A) The evolution of the migration propensity,
(B) the evolution of the maximum migration distance per time
step, (C) the evolution of the expected migration distance, and (D)
the evolution of the number of migratory cells leaving the
neoplasm per time step is unaffected by the initial parameter
settings. Here, the dynamic model with variable lifespan blood
vessels used 100 blood vessels and an occlusion threshold of 20.
Effect of the initial migration propensity and
namics and available resources. In all 4 panels, the
background represents the amount of available resources. White
indicates that there are a lot of available resources, black that there
are none, and blue that there is some small amount. Yellow circles
represent the location of blood vessels. (A) When there are few
blood vessels, but they have long lifespans (e.g. 500 time steps),
natural selection leads to almost complete utilization of input
resources. In this case, only a single recently generated blood vessel
has not yet been completely exploited by the cells. (B) With few
blood vessels that are generated and occluded frequently, the cells
The relationship between blood vessel dy-
Evolution of Cell Migration in Neoplasms
PLoS ONE | www.plosone.org9 April 2011 | Volume 6 | Issue 4 | e17933
do not have enough time to locate and proliferate around a blood
vessel before it disappears. This leads to large quantities of
available resources for any cell that migrates from its current
position, and so there is selection for increased cell migration. (C)
and (D) With a high density of blood vessels, cells are distributed
relatively evenly across space, though at low density for any one
patch, and new blood vessels will likely appear in regions already
occupied by cells that are supported by nearby blood vessels. This
leaves little room for the generation of unutilized resources, though
there are occasional small regions of resources generated when
those blood vessels have short lifespans (D).
migration distance under different rates of evolution
determined by varying the migration propensity muta-
tion standard deviation and the maximum migration
distance mutation standard deviation. Dynamic model
with variable lifespan blood vessels (black line) selects for higher
levels of cell migration in all cases, compared to uniform input of
resources across space and time (red dashed line), or static blood
vessels (blue dotted line). Changing the standard deviation of the
daughter cell migration propensity by a factor of 10 (the ‘‘size’’ of
mutations) has little effect on the expected migration distance,
except in the slope of the initial trajectory in the variable lifespan
blood vessel condition (A, B). However, the standard deviation of
the maximum migration mutation does affect the expected
migration distance (C, D), though, even at the low rate (std.dev.
=0.1; panel C), the expected migration distance continues to
increase and does not reach equilibrium by the end of the 7
Examples of the evolution of the expected
amount of resources required to reproduce. The plots
show sample runs of evolution of migration within neoplasms with
non-uniform cellular proliferation rate under dynamic model with
variable lifespan blood vessels, the uniform input, and the static
blood vessel models. (A) Evolution of the migration propensity. (B)
Evolution of the maximum migration distance per time step. (C)
Evolution of the expected migration distance. (D) The number of
migratory cells leaving the neoplasm per time step. Each panel
shows the results of three different forms of resource supply to the
neoplasm. In this setting, the reproduction threshold is a mutable
phenotype. If a cell gets a new mutation (mutation rate =1022per
cell division), the reproduction threshold of each daughter cell is
Evolution of both migration rate and the
modified by drawing from a truncated normal distribution with
the parental threshold as mean and a standard deviation of 20
units. The initial reproduction threshold is set to be 240 units and
the lowest threshold is 120. The other model parameters remain
the same as in the uniform proliferation rate setting (Figure 1). For
all four outcomes (panels A–D), neoplasms with dynamic blood
vessels still evolve to the highest levels of cell migration with even
higher values of expected migration distance and emigrating cell
numbers. This demonstrates that adding more evolutionary
complexity into our model does not change the fundamental
results. (E) Neoplasms with dynamic blood vessels quickly evolved
increased proliferation ability by lowering the threshold necessary
to reproduce. Since daughter cells receive half of the parent’s
resources at cell division, and cells need to maintain an internal
resource store in order to avoid cell death, there is selection against
setting the reproduction threshold so low that daughter cells would
be on the brink of starvation.
the fixed lifespan blood vessel model. Due to computational
constraints, we have simulated a relatively small neoplasm. To test
how the simulation size might affect the results, we tested models
with 1024, 2116, 4096, 8100, and 16,384 patches and fixed
lifespan blood vessels. The blood vessel number was also scaled
1:41 with neoplasm size (25, 50, 100, 200 and 400) and the lifespan
of blood vessels was set to 6 time steps. The average amount of
available, unutilized resources per patch (A) and the expected
migration distance (B) appear to be approaching an asymptote.
The number of emigrating cells appears to scale linearly as the
neoplasm, and cell population size grows (C).
Scaling the size of the simulated neoplasm in
Parameters and their values used in the
The authors would like to thank Costas Koumenis, Athena Aktipis and
John Pepper for helpful discussions and comments on the manuscript.
Conceived and designed the experiments: JC KS QH CCM. Performed
the experiments: JC. Analyzed the data: JC. Wrote the paper: JC KS QH
CCM. Wrote the software: JC.
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