Background
Non-small cell lung cancer (NSCLC) remains at the top of the list of cancer-related deaths in the United States 1. The epidermal growth factor receptor (EGFR) is frequently overexpressed in NSCLC 23. Binding of epidermal growth factor (EGF) or transforming growth factor alpha (TGFα) to the extracellular domain of EGFR produces a number of downstream effects that affect phenotypic cell behavior including proliferation, invasion, metastasis, and inhibition of apoptosis 4. In particular, increasing the expression of these growth factors leads to EGFR hyperactivity 56, thus increases tumor cell motility and invasiveness, and finally enhances lung metastasis 78. Since approximately 90% of all cancer deaths originate from the spread of primary tumor cells into the surrounding tissue 9, quantitative measurements of the relationship between the level of the growth factors and the resulting tumor expansion is crucial – all the more so, since EGFR has emerged as an attractive therapeutic target for patients with advanced NSCLC 10.
A number of EGFR-related intracellular signal transduction pathways have been studied 111213141516, including NSCLC 17, and corresponding computational models at the molecular-level have been developed. These quantitative works mainly focused on signal-response relationships between the binding of EGF to EGFR and the activation of downstream proteins in the signaling cascade. With these in silico approaches, experimentally testable hypotheses can be made on signaling events controlling divergent cellular responses such as cell proliferation, differentiation, or apoptosis 1819. However, most signaling works did not yet consider the cellular level (see 2021 for a review), and, conversely, only a few recent EGF/EGFR-mediated cellular-level models have started to incorporate a simple molecular level in studying e.g., cell migration in breast cancer 22, cell proliferation 23, and autocrine receptor-ligand dynamics 2425. We argue that a more detailed understanding of a complex cancer system requires integrating both molecular- and cellular-level works to properly examine multicellular dynamics. To our knowledge, to date, no multiscale model of NSCLC has been developed or published.
Our group has been developing multiscale models to investigate highly malignant brain tumors as complex dynamic and self-organizing biosystems. Since this NSCLC model builds on these works, we will briefly review some milestones. First, an agent-based model for studying the spatio-temporal expansion of virtual glioma cells in a two-dimensional (2D) environment was built and the relationship between rapid growth and extensive tissue infiltration was investigated 2627. This 'micro-macro' framework was then extended 'top-down' by incorporating an EGFR molecular interaction network 28 so that molecular dynamics at the protein level could be related to multi-cellular tumor growth patterns 29. Most recently, an explicit cell cycle description was implemented to study in more detail brain tumor growth dynamics in a three-dimensional (3D) context 30. These previous works have provided a computational paradigm in which biological processes have been successfully simulated from the molecular scale up to the cellular level and beyond. This progress led us to test the platform's applicability to and flexibility for other cancer types as well.
In this paper, we have therefore extended these previous modeling works to the case of NSCLC. Necessary modifications include at the molecular level the implementation of a NSCLC-specific EGFR-ERK signal transduction pathway. A novel, data-driven switch that is operated by two key molecules, i.e. phospholipase Cγ (PLCγ) and extracellular signal-regulated kinase (ERK), processes the phenotypic decision at the cellular level. The aim of this in silico work is to provide insights into the externally triggered molecular-level dynamics that govern phenotypic changes and thus impact multicellular patterns in NSCLC. In the following sections, we will first show the detailed design of the model before we present and then discuss the simulation results.
Model
Molecular Signaling Pathway
The kinetic model of the implemented NSCLC-specific molecular signaling pathway, which consists of 20 molecules, is shown in Fig. 1. These proteins, including both receptor (EGFR) and non-receptor kinases (e.g., PLCγ and protein kinase C (PKC) 3132, Raf, mitogen-activated protein kinase kinase (MEK), and ERK 333435), have been experimentally or clinically proven to play an important role in NSCLC tumorigenesis. Although in reality these molecules fulfil their functions by interacting with a multitude of other molecular species from many distinct pathways 3637, we choose to start with these proteins not only because of their significance in the case of NSCLC but also since most of their kinetic parameters can be found in the literature. Also, it is reasonable to reduce the number of involved molecules as a starting point for modeling 38. Amongst these proteins, both PLCγ and ERK are of particular interest for determining the cell's phenotypic changes as we will detail below.
<p>Figure 1</p>
Kinetic model of the NSCLC-specific EGFR signaling pathway
Kinetic model of the NSCLC-specific EGFR signaling pathway. The arrows represent the reactions specified in Tables 1 and 2.
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Kinetic equations are written in terms of concentrations and the reaction rates are functions of concentrations. The association and dissociation steps are characterized by first-order and second-order rate constants, respectively. We note that, although in reality chemical reactions of second or higher order are two-step processes, they are usually treated as a one-step process in mathematical modeling 39. Our model is based on a total of 20 ordinary differential equations (ODEs) and uses exactly the same modeling techniques as other pathway analysis studies (see 1112 for detailed definitions). For simplicity, the ODEs for different molecules were calculated by Eq. (1):
d
(
X
i
)
d
t
=
∑
v
production
−
∑
v
Consumption
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaqcfa4aaSaaaeaacqWGKbazcqGGOaakcqWGybawdaWgaaqaaiabbMgaPbqabaGaeiykaKcabaGaemizaqMaemiDaqhaaOGaeyypa0ZaaabqaeaacqWG2bGDdaWgaaWcbaGaeeiCaaNaeeOCaiNaee4Ba8MaeeizaqMaeeyDauNaee4yamMaeeiDaqNaeeyAaKMaee4Ba8MaeeOBa4gabeaaaeqabeqdcqGHris5aOGaeyOeI0YaaabqaeaacqWG2bGDdaWgaaWcbaGaee4qamKaee4Ba8MaeeOBa4Maee4CamNaeeyDauNaeeyBa0MaeeiCaaNaeeiDaqNaeeyAaKMaee4Ba8MaeeOBa4gabeaaaeqabeqdcqGHris5aaaa@5B71@
where Xi represents one of these 20 molecular pathway components. In Eq. (1), the change in concentration of molecule Xi is the result of the reaction rates producing Xi minus the reaction rates consuming it. Each biochemical reaction is then characterized by vi (see Fig. 1) with forward and reverse rate constants. Tables 1 and 2 summarize the kinetic parameters and the ODEs used for the model.
<p>Table 1</p>
Kinetic equations and initial concentrations. See Table 2 for references.
Reactant
Molecular variable
Initial concentration [nM]
ODE
X
1
EGF
to be varied
d(X1)/dt = -v1
X
2
EGFR
80
d(X2)/dt = -v1
X
3
EGF-EGFR
0
d(X3)/dt = v1 - 2v2
X
4
(EGF-EGFR)2
0
d(X4)/dt = v2 + v4 - v3
X
5
EGF-EGFR-P
0
d(X5)/dt = v3 + v7 - v4 - v5
X
6
PLCγ
10
d(X6)/dt = v8 - v5
X
7
EGF-EGFR-PLCγ
0
d(X7)/dt = v5 - v6
X
8
EGF-EGFR-PLCγ-P
0
d(X8)/dt = v6 - v7
X
9
PLCγ-P
0
d(X9)/dt = v7 - v8 - v9 - v10
X
10
PLCγ-P-I
0
d(X10)/dt = v9
X
11
PKC
10
d(X11)/dt = -v10
X
12
PKC*
0
d(X12)/dt = v10 - v11
X
13
Raf
100
d(X13)/dt = -v11
X
14
Raf*
0
d(X14)/dt = v11 - v12 - v14
X
15
MEK
120
d(X15)/dt = v13 - v12
X
16
MEK-P
0
d(X16)/dt = v12 + v15 - v13 - v14
X
17
MEK-PP
0
d(X17)/dt = v14 - v15 - v16 - v18
X
18
ERK
100
d(X18)/dt = v17 - v16
X
19
ERK-P
0
d(X19)/dt = v16 + v19 - v17 - v18
X
20
ERK-PP
0
d(X20)/dt = v18 - v19
- PKC* and Raf * indicate the activated form of PKC and Raf, respectively.
<p>Table 2</p>
Kinetic parameters. Concentrations and the Michaelis-Menten constants (K4, K8, and K11-K19) are given in [nM]. First- and second-order rate constants are given in [s-1] and [nM-1·s-1], respectively. V4, V8, and V11-V19 are expressed in [nM·s-1].
Reaction number
Equation
Kinetic parameter
Reference
v1
k1·X1·X2 - k-1·X3
k1 = 0.003
k-1 = 0.06
[11]
v2
k2·X3·X3 - k-2·X4
k2 = 0.01
k-2 = 0.1
[11]
v3
k3·X4 - k-3·X5
k3 = 1
k-3 = 0.01
[11]
v4
V4·X5/(K4 + X5)
V4 = 450
K4 = 50
[11]
v5
k5·X5·X6 - k-5·X7
k5 = 0.06
k-5 = 0.2
[11]
v6
k6·X7 - k-6·X8
k6 = 1
k-6 = 0.05
[11]
v7
k7·X8 - k-7·X5·X9
k7 = 0.3
k-7 = 0.006
[11]
v8
V8·X9/(K8 + X9)
V8 = 1
K8 = 100
[11]
v9
k9·X9 - k-9·X10
k9 = 1
k-9 = 0.03
[11]
v10
k10·X9·X11 - k-10·X12
k10 = 0.214
k-10 = 5.25
Estimate
v11
V11·X12·X13/(K11 + X13)
V11 = 4
K11 = 64
[39]
v12
V12·X14·X15/[K12·(1 + X16/K14) + X15]
V12 = 3.5
K12 = 317
[14]
v13
V13·X16/[K13·(1 + X17/K15) + X16]
V13 = 0.058
K13 = 2200
[12]
v14
V14·X14·X16/[K14·(1 + X15/K12) + X16]
V14 = 2.9
K14 = 317
[12]
v15
V15·X17/[K15·(1 + X16/K13) + X17]
V15 = 0.058
K15 = 60
[12]
v16
V16·X17·X18/[K16·(1 + X19/K18) + X18]
V16 = 9.5
K16 = 1.46 × 105
[12]
v17
V17·X19/[K17·(1 + X20/K19) + X19]
V17 = 0.3
K17 = 160
[12]
v18
V18·X17·X19/[K18·(1 + X18/K16) + X19]
V18 = 16
K18 = 1.46 × 105
[12]
v19
V19·X20/[K19·(1 + X19/K17) + X20]
V19 = 0.27
K19 = 60
[12]
Micro-Environment
The 2D virtual micro-environment is made up of a discrete lattice consisting of a grid with 200 × 200 points (Fig. 2). We use p(i, j) to express each point in the lattice, where i and j indicate the integer location in Euclidean terms. One single, distant nutrient source (simulating a cross-sectional blood vessel) is located at p(150, 150). To start with, a number of M × N cells (in other words, an M-by-N matrix) are initialized in the center of the lattice (and this number can be set to meet different simulation purposes). Each grid point can be occupied by one cell only or remain empty at a time.
<p>Figure 2</p>
Two-dimensional virtual micro-environment
Two-dimensional virtual micro-environment. Depicted are the 200 × 200 lattice (left) with the position of the nutrient source, and the seed cells with assignment of the corner cell IDs (0, 6, 42, and 48).
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Three external chemical cues are employed in the model: EGF, glucose and oxygen tension. As we have done in previous studies 2930, the nutrient source carries the highest value of these three diffusive cues, which implicates that it is the most attractive location for the chemotactically acting tumor cells. Then, by means of normal distribution, each grid point of the lattice is assigned a concentration profile of these three cues. The levels of these distributions are weighted by the distance, dij, of a given grid point from the nutrient source. The distributions of these three cues are described by the following equations:
E
G
F
i
j
=
T
m
⋅
exp
(
−
2
d
i
j
2
/
σ
t
2
)
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGaemyrauKaem4raCKaemOray0aaWbaaSqabeaacqWGPbqAcqWGQbGAaaGccqGH9aqpcqWGubavdaWgaaWcbaGaemyBa0gabeaakiabgwSixlGbcwgaLjabcIha4jabcchaWjabcIcaOiabgkHiTiabikdaYiabdsgaKnaaDaaaleaacqWGPbqAcqWGQbGAaeaacqaIYaGmaaGccqGGVaWliiGacqWFdpWCdaqhaaWcbaGaemiDaqhabaGaeGOmaidaaOGaeiykaKcaaa@4AAA@
G
l
u
c
o
s
e
i
j
=
G
a
+
(
G
m
−
G
a
)
⋅
exp
(
−
2
d
i
j
2
/
σ
g
2
)
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGaem4raCKaemiBaWMaemyDauNaem4yamMaem4Ba8Maem4CamNaemyzau2aaWbaaSqabeaacqWGPbqAcqWGQbGAaaGccqGH9aqpcqWGhbWrdaWgaaWcbaGaemyyaegabeaakiabgUcaRiabcIcaOiabdEeahnaaBaaaleaacqWGTbqBaeqaaOGaeyOeI0Iaem4raC0aaSbaaSqaaiabdggaHbqabaGccqGGPaqkcqGHflY1cyGGLbqzcqGG4baEcqGGWbaCcqGGOaakcqGHsislcqaIYaGmcqWGKbazdaqhaaWcbaGaemyAaKMaemOAaOgabaGaeGOmaidaaOGaei4la8ccciGae83Wdm3aa0baaSqaaiabdEgaNbqaaiabikdaYaaakiabcMcaPaaa@594B@
O
x
y
g
e
n
i
j
=
O
a
+
(
O
m
−
O
a
)
⋅
exp
(
−
2
d
i
j
2
/
σ
o
2
)
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGaem4ta8KaemiEaGNaemyEaKNaem4zaCMaemyzauMaemOBa42aaWbaaSqabeaacqWGPbqAcqWGQbGAaaGccqGH9aqpcqWGpbWtdaWgaaWcbaGaemyyaegabeaakiabgUcaRiabcIcaOiabd+eapnaaBaaaleaacqWGTbqBaeqaaOGaeyOeI0Iaem4ta80aaSbaaSqaaiabdggaHbqabaGccqGGPaqkcqGHflY1cyGGLbqzcqGG4baEcqGGWbaCcqGGOaakcqGHsislcqaIYaGmcqWGKbazdaqhaaWcbaGaemyAaKMaemOAaOgabaGaeGOmaidaaOGaei4la8ccciGae83Wdm3aa0baaSqaaiabd+gaVbqaaiabikdaYaaakiabcMcaPaaa@5852@
Moreover, the three chemotactic cues continue to diffuse over the lattice throughout the entire process of a simulation with a fixed rate, using the following equation:
∂
M
i
j
∂
t
=
D
M
⋅
∇
2
M
i
j
,
t
=
1
,
2
,
3
,
...
.
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaqbaeqabeGaaaqaaKqbaoaalaaabaGaeyOaIyRaemyta00aaWbaaeqabaGaemyAaKMaemOAaOgaaaqaaiabgkGi2kabdsha0baakiabg2da9iabdseaenaaBaaaleaacqWGnbqtaeqaaOGaeyyXICTaey4bIe9aaWbaaSqabeaacqaIYaGmaaGccqWGnbqtdaahaaWcbeqaaiabdMgaPjabdQgaQbaakiabcYcaSaqaaiabbsha0jabg2da9iabigdaXiabcYcaSiabikdaYiabcYcaSiabiodaZiabcYcaSiabc6caUiabc6caUiabc6caUiabbccaGiabc6caUaaaaaa@4EAD@
where M represents one of the three external cues, and t represents a time step. The coefficients in Eqs. (2–5) are listed in Table 3 (see also 30 for more details). It is evident then that the closer a given location is to the nutrient source, the higher the levels of the three cues will be at this grid point. Glucose will be continuously taken up by cells to support their metabolism. Only the nutrient source, p(150, 150), is replenished at each time step while all other grid points are not. In addition, cells take up both their own EGF and that secreted by adjoining cells in our model, because cancer cells act in both autocrine and paracrine manner in consuming EGF 4041.
<p>Table 3</p>
Coefficients of distribution and diffusion of EGF, glucose and oxygen tension. Values are taken from the literature 7273.
Coefficient
Value
Units
Description
T
m
2.56
Nm
Maximum concentration of EGF
G
a
17.0
mM
Normal concentration of glucose
G
m
57.0
mM
Maximum concentration of glucose
O
a
0.0017
DC
Normal concentration of oxygen
O
m
0.0025
DC
Maximum concentration of oxygen
D
EGF
6.7 × 10-11
m2·s-1
Diffusion coefficient of EGF
D
Glucose
5.18 × 10-11
m2·s-1
Diffusion coefficient of glucose
D
Oxygen
8.0 × 10-9
m2·s-1
Diffusion coefficient of oxygen
Each cell encompasses a self-maintained molecular interaction network (shown in Fig. 1) and the simulation system records the molecular composite profile at every time step to determine the cell's phenotype for the next step. In between time steps, the chemical environment is being updated, including EGF and glucose concentration as well as oxygen tension (according to Eq. (5)). When the first cell reaches the nutrient source the simulation run is terminated.
Cellular Phenotype Decision
Four tumor cell phenotypes are considered in the model: proliferation, migration, quiescence and death. Cell death is triggered when the on site glucose concentration drops below 8 mM 42. A cell turns quiescent when the on site glucose concentration is between 8 mM and 16 mM, when it does not meet conditions for migration or proliferation (see below), or when it cannot find an empty location to migrate to or proliferate into.
The most important two phenotypic traits for spatio-temporal expansion, i.e. migration and proliferation, are decided by evaluating the dynamics of the following critical intracellular molecules. (1) PLCγ is known to be involved in directing cell movement in response to EGF 434445; PLCγ dynamics are accelerated during migration in cancer cells 46. Therefore, in our model, the rate of change of PLCγ (ROCPLC) decides if a cell proceeds to migration or not. That is, if ROCPLC exceeds a certain set threshold, TPLC, the cell has the potential to migrate. (2) Similarly, the rate of change of ERK (ROCERK) decides if a cell proceeds with proliferation. ERK has been found experimentally to have a strong influence on cell proliferation 334748, and transient activation of ERK with EGF leads to cell replication 4950. If a cell decides to migrate or proliferate, it will search for an appropriate location to move to or for its offspring to reside in. Candidate locations are those grid points surrounding the cell. Implementing a cell surface receptor-mediated chemotactic evaluation, the most appropriate location is detected by using a 'search-precision' mechanism 27 according to:
Tij = ψ·Lij + (1 - ψ)·εij
where Tij represents the perceived attractiveness of location p(i, j), Lij represents the result of an evaluation function for location p(i, j) (see 27 for the definition of Lij), and ε~ N(μ, σ2) is an error term following a normal distribution with mean μ and variance σ2. ψ ∈ [0,1] denotes the search-precision parameter that for a given run is held constant for all cells. Briefly, for a given cell at a certain location, when ψ = 0 the cell performs a pure random walk, whereas when ψ = 1 the cell always selects the location with the highest glucose concentration. Based on previous results 27, we set ψ = 0.7 because this value tends to lead to the highest average velocity of the tumor's spatial expansion.
It is worth noting that even if ROCPLC or ROCERK exceed their corresponding thresholds, it does not necessarily have to lead to cell migration or proliferation. Rather, if nowhere else to go, the cell remains quiescent and continues to search for an empty location at the next time step.
Any cell in the process of changing its phenotype will fall into one of these four categories: (i) ROCPLC < TPLC and ROCERK < TERK; (ii) ROCPLC > TPLC and ROCERK < TERK; (iii) ROCPLC < TPLC and ROCERK > TERK; and (iv) ROCPLC > TPLC and ROCERK > TERK. Figure 3 lists these conditions and their phenotypic consequences, respectively. Following the first three cell decisions is straightforward; first, if a cell experiences condition (i) no phenotypic change results as both ROCPLC and ROCERK remain below their corresponding thresholds; however, if a cell faces condition (ii) the cell migrates because of ROCPLC exceeding its threshold while in the presence of (iii) the cell proliferates due to ROCERK exceeding its threshold. However, for (iv), and in the absence of any specific experimental data, i.e. for the case that both ROCPLC and ROCERK exceed their corresponding thresholds, we explored two hypotheses: 'rule A' yielding migration advantage (i.e., the cell decides to migrate) whereas 'rule B' resulting in a proliferation advantage (i.e., the cell decides to proliferate). For simplicity, decision rules for the first three conditions are referred to 'general rules', while rules A and B are referred to 'special rules' hereafter. In the following section, we will describe the corresponding simulation results.
<p>Figure 3</p>
Cell phenotypic decision algorithm
Cell phenotypic decision algorithm. See text for more details.
![]()
Results
Our algorithm was implemented in C/C++. A total of 49 seed cells were initially set up in the center of the lattice, and these cells were arranged in a 7 × 7 square shape (i.e., M = 7 and N = 7, see Fig. 2 for the configuration of the seed cells). We defined cell IDs from 0 to 48 (left to right, bottom to top). To investigate cell expansion dynamics, we monitored all cells and recorded their molecular profiles at every time step. We are particularly interested in the following four boundary cells: Cell No 0 (bottom-left corner, farthest from the source), Cell No 6 (top-left corner), Cell No 42 (bottom-right corner), and Cell No 48 (top-right corner, closest to the source). Through the distinct micro-environmental conditions they face, these corner cells exemplify the impact of location on single cell behavior, while they however still grasp the nature of the entire system. As described before, both rules A and B were tested for each different simulation condition.
Multi-Cellular Dynamics
Figure 4 shows two simulation results for rules A and B, respectively. The simulations were conducted with a standard EGF concentration of 2.56 nM. Note that this concentration is derived from the literature 5152 and has been rescaled to fit our model as a benchmark starting point for further simulations. In the upper panel of Fig. 4(a) for rule A, tumor cells first display on site proliferation prior to exhibiting extensive migratory behavior towards the nutrient source. However, for rule B (lower panel), cells remain stationary proliferative throughout, thereby increasing the tumor radius yet without substantial mobility-driven spatial expansion. The run time for the latter case (rule B) was considerably longer than for rule A. Based on the criterion chosen for terminating the run, i.e. the first cell reaching the nutrient source, this result is somewhat expected since rule A favors migration whereas rule B promotes proliferation. This is further supported by analysis of the evolution of the various phenotypes and the change of [total] cell numbers (Fig. 4(b)). While both rules generate all three cell phenotypes (proliferation (dark blue), migration (red), and quiescence (green)), rule A (left panel) indeed appears to result in a cancer cell population that exhibits a larger migratory fraction than the one emerging through rule B (right panel) which, however, yields a larger portion of proliferative cells (light blue). It is thus not surprising that for rule B, the [total] cell population of the tumor system exceeds the one achieved through rule A by a factor of 10.
<p>Figure 4</p>
Multicellular tumor expansion dynamics (a)
Multicellular tumor expansion dynamics. (a) Shows the multicellular patterns that emerge through rule A (upper panel) and rule B (lower panel), respectively. (b) Describes the numeric evolution (y-axis) of each cell phenotype as well as of the [total] cell population (light blue) over time (x-axis) for rule A (left panel) and rule B (right panel), respectively. Note: proliferative tumor cells are labeled in dark blue, migratory cells in red, quiescent cells in green and dead cells in grey.
![]()
Influence of Decision Rules on Phenotypic Changes
To better understand the significance of each rule for the tumor system, we have investigated its influence on generating the intended phenotype. Figure 5 shows the weight of rule A on migration (a), and that of rule B on proliferation (b). (The results are taken from the two simulation runs reported in Fig. 4). In Fig. 5(a), migrations derive from two sources: (1) general rule, i.e. [ROCPLC > TPLC and ROCERK < TERK] and (2) rule A; proliferations stem from one source only, i.e. if [ROCPLC < TPLC and ROCERK > TERK]. Rule A plays a more dominant role in triggering migrations than the general rule does, yet does not contribute to increasing proliferations. Likewise, rule B has influence on proliferation only (Fig. 5(b)) and it contributes more to inducing proliferations than the corresponding general rule does.
<p>Figure 5</p>
Weight of decision rules on changing cell phenotypes
Weight of decision rules on changing cell phenotypes. Influence on changing cell migration (left panel) and proliferation (right panel) when following the corresponding rule (see Fig. 3). The dashed red line indicates rule A-mediated migrations in (a), while the dashed blue line denotes rule B-mediated proliferations in (b). Fitting curves in solid black are calculated using a standard linear least squares method. Slopes of the fitting curves are 1.40 cells/step in (a) and 0.03 cells/step in (b), respectively. Note: The drop of the dashed red line in the left panel of (a) is caused by the termination of the simulation when a cell reached the source (in this case, no further computation on remaining cells will be performed).
![]()
However, as documented in the linear least square fittings, the rate at which rule A causes an increase in migration exceeds by far the one by which rule B induces an increase in proliferation. This indicates that the influence of rule A on increasing migrations is more substantial than that of rule B on increasing proliferations. Being particularly interested in gaining insights into spatially aggressive tumors, we continue in the following with investigating the implications of rule A on microscopic and molecular level dynamics of the cancer system.
Phase-Transition at Molecular Level
To further investigate (for rule A) the relationship between EGF concentration and phenotypic changes we varied the extrinsic EGF concentration from the standard value of 2.65 × 1.0 nM to 2.65 × 50.0 nM by an incremental increase of 0.1 nM in each simulation. As a result of the model's underlying chemotactic search paradigm, expectedly a simulation under the condition of a higher extrinsic EGF concentration finished faster than that with a lower one. However, cells turn out not to exhibit completely homogeneous behavior.
Specifically, we focus on Cell No 48, the cell closest to the nutrient source, and report its corresponding molecular changes in Fig. 6. One can see that as the standard EGF concentration increases, the number of proliferations (blue) decreases gradually up to a phase transition between 2.65 × 31.1 and 2.65 × 31.2 nM. That is, if the standard EGF concentration is less than 2.65 × 31.1 nM, proliferation still occurs in this particular cell, but if the ligand concentration starts to exceed 2.65 × 31.2 nM, its proliferative trait entirely disappears. In the presence of nutrient abundance, a very minor increase in extrinsic EGF can apparently abolish the expression of a phenotype. Even more intriguing, although the subcellular concentration change appears to be rather similar with regards to its patterns, on a closer look, the peak maxima of the rate changes for PLCγ and the turning point of the rate changes for ERK occur at an earlier time point for increasing EGF concentrations. This finding suggests that in the presence of excess ligand, the here implemented intracellular network switches to a more efficient signal processing mode. We note that for cell IDs 0, 6, and 42, no such phase transition emerged (data not shown) hence further supporting that this behavior is concentration dependent, and that geography, i.e. a cell's position relative to nutrient abundance, matters. Confirming the robustness of our finding for Cell No 48 we note that this cell continued to experience a phase transition when the coordinates of the center of the initial 49 cells was set randomly within a square region where p(100,100) is the lower left corner and p(110,110) is the upper right corner (5 runs, data not shown).
<p>Figure 6</p>
Changes at the molecular level for Cell No 48 with an increasing extrinsic EGF concentration (rule A)
Changes at the molecular level for Cell No 48 with an increasing extrinsic EGF concentration (rule A). Four simulation runs are depicted where (from left to right) the EGF concentration increases from 2.65 × 1.0 to 2.65 × 31.1, 2.65 × 31.2, and finally, to 2.65 × 50.0 nM. (From top to bottom) plotted are the absolute change of PLCγ, rate of change of PLCγ, and rate of change of ERK. Note that the number of proliferations is decreasing gradually and finally disappears at a phase transition between the EGF concentrations of 2.65 × 31.1 and 2.65 × 31.2 nM. (For phenotype labeling see Fig. 4).
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Discussion & Future Works
While using mathematical models to investigate the behavior of signaling networks is hardly new, understanding a complex biosystem, such as a tumor, by focusing on the analysis of its molecular or cellular level separately or exclusively is insufficient, particularly if it excludes the interaction with the surrounding tissue. Recent analyses of signaling pathways in mammalian systems have revealed that highly connected sub-cellular networks generate signals in a context dependent manner 53. That is, biological processes take place in heterogeneous and highly structured environments 54 and such extrinsic conditions alone can induce the transformation of cells independent of genetic mutations as has been shown for the case of melanoma 55. Taken together, modeling of cancer systems requires the analysis and use of signaling pathways in a simulated cancer environment (context) across different spatial-temporal scales.
Our group has been focusing on the development of such multiscale models for studying highly malignant brain tumors 27293056. Here, on the basis of these previous works, we presented a 2D multiscale agent-based model to simulate NSCLC. Specifically, we monitored how, dependent on microenvironmental stimuli, molecular profiles dynamically change, and how they affect a single NSCLC cell's phenotype and, eventually, the resulting multicellular patterns.
Proceeding top-down in our analysis, we first evaluated the multicellular readout of molecular 'decision' rules A and B (versus general rules; Fig. 3). The patterns of a more stationary, concentrically growing cancer system (following rule B) are quite different from the rapid, chemotactically-guided, spatial expansion that can be seen in the tumor regulated by rule A (Fig. 4(a)). Not surprisingly, the latter also operates with many more migratory albeit overall less [total] cells (Fig. 4(b)). Furthermore, examining in more detail the influence of the two distinct rules on their respective phenotypic yield, we found that the impact of rule A on increasing cell migration is more substantial than rule B's influence on furthering proliferation (Fig. 5). This finding suggests that the migratory rule A can operate the cancer system through incrementally smaller changes (while the simulation system is more robust for rule B). Such sensitivity to migratory cues corresponds well with experimental data on the response of human breast cancer cells, which showed that a spatially successful expansive system reacts rather quickly to even miniscule changes in chemotactic directionality 5758.
Continuing therefore with rule A, our effort was then geared to gain insights into tumor expansion dynamics not only with regards to extrinsic stimuli but also to cell geography, i.e. a cell's location relative to the replenished nutrient source. Most interestingly, we found a phase transition in the cancer cell closest to the nutrient source (i.e. Cell No 48, while none of the other three corner cells showed similar behavior). Specifically, for a tumor cell at this location, i.e., facing nutrient abundance, proliferation is completely abolished once the extrinsic EGF concentration exceeds a certain level. While this at first may seems rather unexpected, this finding however only confirms the experimentally sound notion that EGF stimulates the spatial expansion of a cancer system 5678. Moreover, with increasing EGF concentrations, the maxima of ROCPLC (Fig. 6) gradually occur earlier which seems to indicate that, under these conditions, the downstream signal is processed faster. Interestingly, such a 'no proliferation, just migration' behavior in the presence of chemo-attractant has indeed already been reported in several in vitro studies using a variety of cancer cell lines 5960 as well as in non-cancerous human cells 61. (While admittedly, for the reasons stated, rule B did not receive similar attention in our analysis), we nonetheless argue that, on the basis of our results and the experimental reports they seem to correspond with, rule A and thus a migratory decision prompted by a [ROCPLC > TPLC and ROCERK > TERK] condition is a reasonable outcome for the signaling process taking place in NSCLC also in vitro and in vivo.
However, moving the model closer to reality will require a multitude of adjustments, one of which is its ability to account for up- or down-regulation in key molecules as a result of tumorigenesis. As a first step, and since experimental data on over-expression of EGFR in a variety of cancer types, including NSCLC, are ample 62636465 we have begun to simulate the impact of an increasing number of receptors on the cancer system (Fig. 7; simulations conducted with an EGFR concentration of 800 nM (per system)). Comparing this preliminary data with those reported in Fig. 6 (simulations conducted with an EGFR concentration of 80 nM (per system)), we find that an EGFR-overexpressing NSCLC tumor seems to operate with even more migration and does so earlier on. The result is a spatially even more aggressive cancer system, which seems to correspond well with the aforementioned experimental studies. And, intriguingly, while the phase transition itself is preserved, it however occurs already at a smaller EGF concentration, hence indicating that the increase in receptor density leads to an amplification of the downstream signal, which again corresponds well with experimental results in examining signaling activities generated by different EGFR family members 66. Taken together, while preliminary, this finding demonstrates applicability and confirms flexibility of this multiscale platform, hence warrants its further expansion.
<p>Figure 7</p>
Changes at the molecular level for Cell No 48 with an increasing extrinsic EGF concentration (rule A), at an EGFR concentration of 800 nM
Changes at the molecular level for Cell No 48 with an increasing extrinsic EGF concentration (rule A), at an EGFR concentration of 800 nM. Three simulation runs are depicted where (from left to right) the EGF concentration increases from 2.65 × 1.0 to 2.65 × 5.9 and 2.65 × 6.0 nM. (From top to bottom) plotted are the absolute change of PLCγ, rate of change of PLCγ, and rate of change of ERK. Note that a phase transition emerges again between the EGF concentrations of 2.65 × 5.9 and 2.65 × 6.0 nM, hence at a lower concentration compared to the one depicted in Fig. 6 (EGFR concentration of 80 nM). In the two simulations around the phase transition, the maximum rates of change for both PLCγ and ERK (i.e., ROCPLC and ROCERK at 2.65 × 5.9 and at 2.65 × 6.0 nM) are lower compared with those in Fig. 6 (i.e., ROCPLC and ROCERK at 2.65 × 31.1 and at 2.65 × 31.2 nM). (For phenotype labeling see Fig. 4).
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There are a number of research tracks that can and should be pursued in future works. First, it will be intriguing to see if, in the presence of a non-replenished nutrient source, the proliferative phenotype eventually can be recovered once extrinsic ligand concentrations fall beyond the phase-transition threshold. More generally, while most of the pathway's parameters, including rate constants and initial component concentrations were obtained from the experimental literature, this data naturally originated from a variety of often stationary experimental settings and different cell types. It therefore represents a less desirable and reliable input than time series data that come from one experimental setting only. Also, some parameters had to be estimated, much like in other well-established pathway models 1112. Taken together, future works will have to include not only proper experimental verification of the estimated parameters and evaluation of the simulation results but also, on the in silico side, techniques such as sensitivity analysis to help determine the effects of parameter uncertainties on model outcome 67 and to identify control points for experiment design 68. While a pathway model cannot be a biological representation in every detail 38 we plan on adding, in incremental steps, other pathways of relevance for NSCLC such as e.g. PI3K/PTEN/AKT 69. Moreover, simulating a more heterogeneous biochemical environment and implementing both cell-cell and cell-matrix interactions 70 are planned steps at the cellular level that should help representing the cancer system of interest in more detail.
Regardless, we believe that the current model already provides useful insights into NSCLC from a systematic view in terms of quantitatively understanding the relationship between extrinsic chemotactic stimuli, the underlying properties of signaling networks, and the cellular biological responses they trigger. Our results yield several experimentally testable hypotheses and thus further support the use of multiscale models in interdisciplinary cancer research. To our knowledge, this presents the first multiscale computational model of Non-Small Cell Lung Cancer and is thus potentially a significant first step towards realizing a fully validated in silico model for this devastating disease.