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Cancer as robust intrinsic state shaped by evolution: a key issues review

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2017 Rep. Prog. Phys. 80 042701

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Cancer as robust intrinsic state shaped by evolution: a key issues review

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1. Introduction

Cancer is a serious and often fatal disease that features malig-nant tumor invasion and metastasis. It has been a great chal-lenge for both medical practitioners and scientists [1–4].

Despite the extended life expectancy, the continual increase of cancer incidences and the unsatisfactory decrease of the asso-ciated mortality rates raise serious concern on the prevailing understanding of cancer mechanism as well as clinical treat-ments [5]. Numerous hypotheses were formulated on cancer

Cancer as robust intrinsic state shaped by evolution: a key issues review

Ruoshi Yuan1, Xiaomei Zhu2, Gaowei Wang1, Site Li1 and Ping Ao1,2

1 Key Laboratory of Systems Biomedicine, Ministry of Education, Shanghai Center for Systems Biomedicine, Shanghai Jiao Tong University, Shanghai 200240, People’s Republic of China2 Shanghai Center of Quantitative Life Sciences, Shanghai University, Shanghai 200444, People’s Republic of China

E-mail: aoping@sjtu.edu.cn

Received 31 July 2015, revised 27 October 2016Accepted for publication 13 December 2016Published 17 February 2017

Corresponding Editor Professor Robert H Austin

AbstractCancer is a complex disease: its pathology cannot be properly understood in terms of independent players—genes, proteins, molecular pathways, or their simple combinations. This is similar to many-body physics of a condensed phase that many important properties are not determined by a single atom or molecule. The rapidly accumulating large ‘omics’ data also require a new mechanistic and global underpinning to organize for rationalizing cancer complexity. A unifying and quantitative theory was proposed by some of the present authors that cancer is a robust state formed by the endogenous molecular–cellular network, which is evolutionarily built for the developmental processes and physiological functions. Cancer state is not optimized for the whole organism. The discovery of crucial players in cancer, together with their developmental and physiological roles, in turn, suggests the existence of a hierarchical structure within molecular biology systems. Such a structure enables a decision network to be constructed from experimental knowledge. By examining the nonlinear stochastic dynamics of the network, robust states corresponding to normal physiological and abnormal pathological phenotypes, including cancer, emerge naturally. The nonlinear dynamical model of the network leads to a more encompassing understanding than the prevailing linear-additive thinking in cancer research. So far, this theory has been applied to prostate, hepatocellular, gastric cancers and acute promyelocytic leukemia with initial success. It may offer an example of carrying physics inquiring spirit beyond its traditional domain: while quantitative approaches can address individual cases, however there must be general rules/laws to be discovered in biology and medicine.

Keywords: cancer, endogenous molecular–cellular network, systems biology, stochastic differential equation, potential landscape, non-equilibrium processes, Darwinian dynamics

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Key Issues Review

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along its research path. One of the earliest, known as the seed and soil hypothesis [6], was formed during the second half of the 19th century. It concisely captured the solid clinical observations that the occurrence of cancers in tissues is highly non-uniform and the migration or metastasis of cancers is also not arbitrary. All humans develop from a single fertilized egg and the genomes of cells in a body are statistically the same. This hypothesis thus indicates that additional factors other than random mutations, in a modern perspective, are involved in cancer genesis and metastasis. On the other hand, after the discovery of DNA and the postulation of the central dogma, cancer has been considered a disease of the genome; this viewpoint is still prevailing today [7]. Meanwhile, increas-ing evidence such as cancer patients without mutations [8, 9], patients under long-term remission with carcinogenic muta-tions [10], and spontaneous regression of cancer [11], suggest the necessity of thinking outside of the mutation box [9, 12].

From the two-hit hypothesis [13, 14] to oncogene coopera-tion [7, 15], and from a genetic disease [7, 16] to incorporating the microenvironment [17, 18], the paradigm of cancer research has been evolving with a trend towards system considerations. Such a trend becomes more evident following the Human Genome Project [19, 20] when large ‘omics’ datasets are read-ily available. It is now acknowledged that a few isolated genes are not sufficient to understand cancer. This observation leads cancer research to another extreme of its linear extension, to pursue ‘larger’ and ‘more complete’ statistical models. These models are inferred from high-throughput data and may not be useful for practical purposes such as predicting new cancer drugs [21]. One may always have to work with problems arising from insufficiency (with respect to the enormous model size) [22] and inaccuracy of high-throughput data [23], which cannot be improved in the foreseeable future. Rather, we demonstrate that there may still be a conceptually simple and practically useful approach to understand cancer by seeking the dynamical organization principle of the underlying molecular interactions.

We proposed in 2007 [3] that cancer corresponds to emer-gent robust state(s) from interactions among endogenous mol-ecules. The endogenous molecular–cellular network is built up evolutionarily for developmental processes (differentia-tion, angiogenesis, etc) and physiological functions (cell cycle, apoptosis, immune response, metabolism, cell adhesion, etc). Carcinogenesis may be understood as a transition from a nor-mal state to cancer state(s) in a potential landscape, as shown intuitively in figure 1. Cancer is thus hypothesized to be formed by evolution but not optimized for the interest of the whole organism. The potential landscape is a quantitative realiza-tion of pioneering ideas: Wright’s adaptive landscape [24] and Waddington’s developmental landscape [25]. Continual efforts have been made to develop these appealing concepts mathe-matically [26–34]. In our framework, the potential landscape is obtained through a decomposition of the Darwinian evolution dynamics of the endogenous molecular–cellular network [28] and robust states correspond to the valleys in the landscape. Previously, cancer has been considered as robust pathological homeostasis state(s) [35] and hypothesized as robust state(s) from different mechanistic viewpoints, which are consistent with our theory [3, 31, 36–41].

The endogenous network hypothesis for cancer has recently been tested in prostate [42], hepatocellular [43], gastric can-cers [44], and acute promyelocytic leukemia [45]. Major difficulties encountered in cancer research, such as cancer het-erogeneity [4, 44], drug resistance [46, 47], patients without mutations [8, 9] and patients under long-term remission with carcinogenic mutations [10], spontaneous regression of can-cer [11], are naturally resolved (at least conceptually). These models also lead to new predictions, such as new biomarkers and drug targets, which may be verified in the future by new experiments [42–44].

Previous studies on model organisms such as Phage lambda [48] demonstrate that a relatively few key endogenous agents (e.g. CI and Cro) are sufficient to characterize major behaviors of a biological system [49–51]. We argued that the top level of the decision making, the core endogenous molecular–cellular network, is composed of several modules with relative auton-omous functions [50, 52, 53]. With a careful invest igation on the necessary modules, we evaluated, selected, and consoli-dated key agents responsible for each module and interactions between them based on solid biochemistry and molecular biology experiments [3, 42–44, 54].

We found that robustness of a biological system could be acquired if the underlying bionetwork has a topological structure so that a few collective modes, or states, represent-ing physiological functions or pathological conditions exist, much in the similar situations encountered in many-body physics. From the landscape viewpoint, this robustness is manifested by a few large basins and transition barriers from one basin to another [55]. In a detailed study of acute pro-myelocytic leukemia (APL) [45], as shown in figure 13, we demonstrate that the ‘collectiveness’ of the network are real-ized by positive feedback among a large number of effective nodes and a common inhibition to other positively reinforced clusters.

A major obstacle in modeling biological networks is the lack of detail on interactions between network nodes. For instance, there is a large number of unknown in vivo param-eters [56, 57] in a dynamical model of a metabolic network using chemical rate equations. In order to overcome such a difficulty, we employed a coarse-grained description of the network dynamics that considers each interaction from one agent to another as ‘activation’ or ‘inhibition’. We used Boolean dynamics as well as a nonlinear and coupled sto-chastic differ ential equation  (SDE) to simulate the network dynamics. Consistent to our assumption, the computed stable states are insensitive to interaction details at the core network level, as shown by random parameter tests (figure 10) and per-turbations on the dynamical structure [42–44]. These stable states with a relatively large basin of attraction are recognized as robust states. A flow diagram of the modeling process is summarized in figure 2.

In this review, we discuss the endogenous network mod-eling with prostate cancer [42] as a main example. The core endogenous network consists of 7 modules, 38 agents, and 115 interactions, as shown in figure 3. We found 10 dynami-cally robust states together with 16 transition states (fixed point with one unstable eigenvalue, i.e. with positive real part,

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of its Jacobian matrix) and 13 hyper-transition states (fixed point with more than one unstable eigenvalues of its Jacobian matrix) by solving differential equations. One of the robust states that had elevated proliferating and immune activities, high concentration of growth factors, inhibited apoptosis, as well as specific metabolic and adhesive behaviors was identi-fied as the prostate cancer state. The assignment was validated by comparison with high throughput data. The potential land-scape with robust states and unstable states connecting them is capable of explaining carcinogenesis and predicts multiple paths of such malignant transitions that contribute to the phe-notypic heterogeneity [4, 44].

The review is organized as follows: in section  2, we first reason for the existence of the endogenous molecular–cellular network formed by evolution. The idea is obtained

from our previous study on Phage lambda genetic switch. In the following part of section 2, we focus on the nonlinear SDE modeling of the endogenous network and supplement Boolean dynamics as reference—cancer and normal physi-ology are emergent robust states. A key correspondence between stochastic and deterministic dynamics involved in using SDEs is emphasized, such that the obtained land-scape is insensitive to ‘temperature’, the noise strength [58]. Endogenous network modeling for prostate cancer is demonstrated with details in section 3. Modeling results for hepatocellular and gastric cancers as well as acute promye-locytic leukemia are also briefly reviewed. Discussion on a systems view of cancer and pharmaceutical application of our framework are in section 4. We list a few open problems in section 5.

Figure 1. Potential landscape for endogenous molecular–cellular network with healthy, cancer, and other robust biological states: three typical situations. The landscape emerges from nonlinear interactions within the endogenous network, its crosstalk with other levels, and the influence of noise. Cancer genesis may be understood as a stochastic transition from a healthy state to a cancer state through a transition state, or a group of transition states. Perturbations such as environmental change may lead to a reshape of the potential landscape, e.g. from a to b, the healthy state is from stable to metastable, resulting an increase of the transition probability to cancer states. For patients at a terminal stage, the landscape may display as c, showing the healthy state becomes critical or even unstable. From the view of phase transition in physics, the multiple robust states would denote the different phases of an organism (matter). Therefore the genesis and development of cancer would be considered as phase transitions. Adapted from [3]. Copyright (2008), from Elsevier.

AgentsModules &Pathways

NetworkDynamicalModeling

Validation

Boolean dynamics

Ordinary differential equations (ODEs)

Stochastic differential equations (SDEs)

A-type Integration

Low throughput data(RT-qPCR, IHC, WB,

SB, FISH)

High throughput data(Microarray, omics-data)

Evolutionarily conserved

Normal physiological condition

Closed by feedbacks &Autonomous system

Biological knowledgefrom literature verified by

biochemistry & molecular biology experiments

Direct interaction between agents:transcriptional regulation

(de)phosporylationcomplex forming, etc.

Essential functionalmodules & pathways such as Cell Cycle,

Apoptosis, Metabolism &

Stress Response,Immune Response, etc.

Interactions

Prediction &Application

The acceptancecondition is

restricted to the reliability &

accuracyof experimental data

High-throughputdata: 70%

Low-throughputdata: 95%

Module level:100%

In silico search of drug targets and

biomarkers

New predictions tobe tested experimentally

Self-consistancy, e.g. random parameter test

Figure 2. Flow diagram for endogenous molecular–cellular modeling of cancer. The central endogenous network is constructed based on a hierarchical structure in the regulatory network of a cell and feedback from its surrounding environment. We first select important modules responsible for basic cell functions as well as those related to cancer and then identify key agents for each module. After agents are selected and consolidated, we perform a large amount of literature search for the interaction between agents with solid experimental confirmation and build the network. Note that the network construction is an iterative process. The acceptance condition for a network would be: for high-throughput data that are completely different type from those used to construct the model, e.g. microarray data, the consistency rate should be about 70%, considering the quality of the data; for low-throughput data that are used to build the model, the rate should be above 95% for self-consistency; and at modular level, the rate should be 100%. Reproduced from [44]. CC BY 3.0.

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2. Endogenous molecular–cellular network: existence, construction, and quantification

2.1. Endogenous molecular–cellular network hypothesis

We start by summarizing the endogenous molecular–cellular network hypothesis. The molecular–cellular agents and their interactions that are built up evolutionarily for maintaining normal physiological functions and developmental processes, are also responsible for the emergence of cancer. By unitiz-ing the hierarchical structure inherited in biological systems, a group of ‘decision-making’ agents can be selected and consolidated composing the endogenous network. Feedback loops are needed for the network to be closed: no absolute upstream and downstream agents. The state of the network may be specified by the expression or activity levels of the endogenous agents. The time evolution of the network state forms a high dimensional stochastic and nonlinear dynamical system. This dynamics is autonomous. It can generate many locally stable states with obvious or non-obvious biological functions. Stable states that have a sufficiently large attrac-tive basin and are insensitive to structural perturbations are identified as robust states. Cancer and normal functions corre-spond to different intrinsic robust states. Carcinogenesis may be understood as a transition from the basin of a normal state to that of a cancer state. Figure 4 summarizes how the ‘closed’

endogenous network responses to input, makes decisions, and generates output. We discuss more detail of the modeling in the following sections.

2.2. Key molecules: lessons by the Phage lambda genetic switch and other systems

Bacteriophage lambda is a virus that infects Escherichia coli (E. coli) cells. As one of the simplest living organism, the genome of phage contains about 40 genes. It has been well studied and experimental knowledge is available at all levels ranging from physics, chemistry to genomics, proteomics and more. Almost every component of it is known in great detail. The most attractive phenomenon about Phage lambda is its ability to switch lifestyles: inside a host bacterium, it chooses one of the two modes of growth, lytic or lysogenic. Phage lambda can live with the host cell as dormant residency in the lysogenic state. Or it can use molecular-genetic apparatus of the host cell for executing its own ontogenetic program to produce new phage particles and cleavage the host cell. It is observed that the ‘switch’ between these two states are both highly stable and highly efficient. While phage can grow in the lysogenic state for millions of generations, it changes to the lytic state at a rate of almost 100% when exposed to a spe-cific environmental signal. Different sets of phage genes are

Figure 3. Endogenous molecular–cellular network for prostate cancer. This core network consists of 7 modules, 38 nodes, and 115 interactions. Red lines with dots denote inhibition and green lines with arrows represent activation. Each interaction is based on solid biochemistry and molecular biology experiment found in literature. Note that in the hierarchical modeling of prostate cancer [42], the prostate specific module is further characterized by a prostate specific sub-network with 60 nodes.

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expressed in these two states through complex interactions, but an autonomous endogenous network with only two key agents CI and Cro were found to be able to explain exper-imental data quantitatively [49–51, 59]. Briefly speaking, if CI occupies the operator, it blocks the synthesis of Cro and also activates its own expression and vice versa. These two ‘master’ regulatory agents control various downstream genes, leading to the lytic or lysogenic state separately. Similar key agents have also been identified in other organisms, e.g. the SinR, SlrR, and SinI proteins in deciding sessile-multicellular or motile-solitary lifestyles of Bacillus subtilis [60] as well as the gene cluster kaiABC in controlling circadian oscilla-tion of cyanobacteria [61]. These studies clearly demonstrate the inherent hierarchy in biological systems, which we will explore with more details in the next section.

2.3. Hierarchical organization: molecules, modules, and networks

It has been argued at the end of the 20th century the changing paradigm of biological research from individual molecules to functional modules [52]. We further argued that the crosstalk between functional modules forms the endogenous molecu-lar–cellular network [50]. By definition, a functional module is a separable and relatively autonomous component of a bio-logical system. For example, cell cycles can be considered as

a functional module, it is both evolutionarily conserved and its function of self-replication is a basic characteristic of life. Apoptosis is another critical module for multi-cellular organism to remove damaged cells and sustain homeostasis. Several onco-gene repressors, such as p53 [62, 63] are among main players in the crosstalk between cell cycle and apoptosis modules [64, 65]. We include several functional modules in modeling cancer: cell cycle, apoptosis, immune response, metabolism, growth fac-tors, tissue specific module etc, for intracellular regulation, and extracellular matrix module for incorporating microenviron-ment [66]. We then identified key agents in each module similar to CI and Cro in Phage lambda genetic switch. The interactions of these molecules are searched from the literature one by one with convincing molecular biological experiments. Feedback from the microenvironment involving cell adhesion and regu-latory molecules such as hormones are included to construct a dynamically autonomous network. Overall, the endogenous molecular–cellular network for cancer is built by utilizing the inherent hierarchical organization of a living organism.

2.4. Mathematical equations based on endogenous molecular–cellular network

Endogenous molecular–cellular network is hypothesized to form during the long-time evolution process [3]. We have proposed a quantitative description of the evolution process

Figure 4. Input and output of the ‘closed’ decision-making endogenous network. Endogenous molecular–cellular network is a ‘closed’ network that can generate multiple dynamically robust states, each robust state represents a specific decision of the cell. Decision-making can be influenced by the changes in the environment through signaling transductions that eventually take effect on values of the agents or by spontaneous transitions between robust states due to intrinsic noise accumulation. Decisions are executed by the downstream targets of the network nodes.

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through a framework on stochastic differential equations [28]. Such stochastic Darwinian dynamics is argued to be univer-sally applicable in describing biological processes and is used for the dynamical modeling of the endogenous molecular–cellular network. Another hurdle would be the inevitable uncertainties [23] involved in modeling complex biological networks, e.g. the lack of interaction details between network nodes, such as in vivo parameters. A working approach is to employ a coarse-grained description of the network dynam-ics that abstracts each interaction from one agent to another as ‘activation’ or ‘inhibition’. More complicated logic may be resolved by adding auxiliary nodes and by further refined descriptions. Here, the quantitative simulation of the network dynamics is through both nonlinear stochastic differential equations and Boolean dynamics.

2.4.1. Stochastic differential equations. In comparison with sketchy Boolean dynamics [67] or with computation-ally expensive chemical master equations (CMEs) [68] of full details, SDE modeling is at a ‘mesoscopic’ level both in the scale of the model (e.g. the number of variables and parameters) and in the biological phenomena to describe—we do not expect to recapitulate detailed behaviors, but globally emergent properties from limited biological knowledge. The general form of the stochastic differential equation for the expression or activity levels of the n agents in the network ( )= …x x xx , , , n

T1 2 can be written for each xi,

= …i n1, , :

( ) ( )α ζ= +x

tf tx x

d

d, ,i

i i (1)

where ( )αf x,i is a nonlinear function with parameters α, and ζi is the multiplicative Gaussian white noise with zero mean

and ( ) ( ) ( ) ( )ζ ζ δ= −′ ′εt t D t tx x x, , 2i j ij . Here ε denotes the

noise strength and ⋅ is the average over the noise distribution, ( )D xij is the ith-row and jth-column element of the diffusion

matrix ( )D x , and δ represents the Dirac delta function. When ε goes to zero, equation (1) reduces to an ordinary differential equation (ODE). A specific realization of fi would be to write it as a sum of generation and degradation of agent i

( ) ( )ατζ= − +

x

tg

xtx x

d

d, ,i

ii

ii (2)

where ( )αg x,i is a nonlinear function simulating activation and inhibition of other agents on the generation of agent i, and τi is the degradation time for agent i. We further simplify the model by normalizing all τ = 1i , that is to assume the domi-nant degradation of agents is caused by cell proliferation. And we use the Hill equation type ( )αg x,i [54, 69] in the modeling

=∑ ⋅+∑ ⋅

⋅+ ∑ ⋅

−fa x

a x a xx

1

1

1iu iu u

n

u iu un

v iv vn i

iu

iu iv (3)

where n is the Hill coefficient and a is the inverse of the appar-ent dissociation constant, ∑u and ∑v mean summation over the sets of all agents activating and inhibiting agent i sepa-rately. An example of constructing Boolean dynamics and differ ential equation for the agent Myc is shown in figure 5.

2.4.2. Boolean dynamics. Boolean dynamics is a useful tool [67, 70–72] to obtain architectural information rather than details about a network. The modeling results were shown in a series of studies to match the regulatory patterns observed in experiments [67, 71, 72]. We apply this most simplified way of modeling network dynamics as a reference to SDE mod-eling. The definition is shown in figure  5 using Myc as an example:

( ) ( ( )    ( )    ( )) (  ( ( )    ( )))β

+ =−

Myc t pRb t E F t MAPK tp t TGF t

1 OR 2 OR ANDNOT 53 OR

where each state variable Myc, pRb, etc, has a discrete value 0 or 1 representing low/high expression separately; t denotes the current state of the variables and t + 1 for the next state; NOT, AND, and OR are the usual logic operators. An equiva-lent formulation with threshold function can also be provided:

( )( )

( ) [ ( )]   [ ]⩽

⎪

⎪⎧⎨⎩

⎧⎨⎩

∑+ =

>

+ = =>

S ta S t

t t t txx

S AS x

11 0

0 otherwise

1 ,1 00 0

i jij j

ii

i

where   = …S i n, 1, ,i means the state values for the n agents in the network; aij = −N for inhibition from agent j to agent i and aij = 1 for activation from j to i; S are the vector form of Si and A the matrix form of aij; ( )t x is a vector-valued threshold function.

Figure 5. From network to dynamics. The definition of Boolean dynamics and differential equations used in this review are illustrated for the agent Myc in figure 3. The task of searching local maxima of the steady state distribution described by a stochastic differential equation system can reduce to solving stable states of the deterministic part differential equations within the framework of A-type stochastic integration. Thus the explicit dependency on noise is omitted here. Applying traditional Ito or Stratonovich interpretations of a stochastic differential equation will break such correspondence between deterministic and stochastic dynamics even for constant diffusion. Note that both Boolean dynamics and differential equations can be implemented by various forms, our modeling results can be stable under such variations.

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Boolean dynamics can be directly generalized to allow multiple values for an agent’s state or to a model with con-tinuous time, but its discrete nature makes it hard to simulate small internal fluctuations [67] that are inherent in biological systems [73]. As we will demonstrate later, stochastic trans-itions by the accumulation of small internal noise are critical in cancer genesis and development. For such a purpose, stochas-tic differential equations are preferred in our framework. Key additional information such as transition states (saddle points) can be explicitly obtained. Comparing to chemical master equations, SDE is computationally more efficient and is shown to be consistent with CME when molecule numbers are not extremely small [74], the usual case for endogenous agents.

2.5. ODE and SDE: A-type stochastic integration leading to robust landscape with respect to noise

The genesis and development of complex diseases such as cancer are typical non-equilibrium processes from a physical viewpoint [75]. We have developed a general framework for studying non-equilibrium processes employing nonlinear sto-chastic differential equations (a generalized form of Langevin equations in physics) [27, 28, 76], through a decomposition of the original stochastic dynamics equation (1) into three comp-onents—a friction matrix ( )S x , an antisymmetric transverse matrix ( )A x , and a potential function (landscape) ( )αφ x, ,

[ ( ) ( )] ( ) ( )∑αφ

ξ+ = −∂∂

+S Ax

t xtx x

xx

d

d

,,

jij ij

j

ii (4)

where ξ is zero mean white noise and satisfies a similar rela-

tion ( ) ( ) ( ) ( )ξ ξ δ= −′ ′εt t S t tx x x, , 2i j ij . In transforming

equation (1) to equation (4), the separate correspondence of the deterministic part and the stochastic part generates two relations [76]: The stochastic part leads to a generalized Einstein relation,

[ ( ) ( )] ( )[ ( ) ( )] ( ) ∑ + − =S A D S A Sx x x x x xs t

is is st tj tj ij,

for each ⩾i j providing n(n + 1)/2 equations; the determinis-tic part

[ ( ) ( )] ( ) ( )∑ α αφ+ = −∂S A fx x x x, , ,j

ij ij j i (5)

with the zero curl of the potential function, we obtain

{[ ( ) ( )] ( )} {[ ( )

( )] ( )}

∑ ∑α

α

∂ + − ∂

+ =

S A f S

A f

x x x x

x x

,

, 0,

is

js js s jt

it

it t

for i > j giving n(n − 1)/2 equations. These n2 equations under proper boundary conditions can determine the n2 unknowns in

( ) ( )+S Ax x . Then the potential function can be calculated by integrating [ ( ) ( )] ( )α+S Ax x f x, . The corresponding Fokker–Planck equation for equation (4) is [76]

( ) [ ( ) ( )][ ( ) ] ( )∑ αρ φ ρ∂ = ∂ + ∂ + ∂εt D Q tx x x x x, , ,ti j

i ij ij j j,

where ( )Q x is an antisymmetric matrix determined by the rela-tion [ ( ) ( )] [ ( ) ( )]+ = + −D Q S Ax x x x 1. We thus connect the stationary probability density function ( )ρ x in the phase space with the potential function ( )αφ x, ,

( ) ( )⎡⎣⎢

⎤⎦⎥

αρ

φ∝ −

εx

xexp

,. (6)

Note that the matrices S, A, D, and Q may also depend on α. When →ε 0, / =x t fd di i, thus

( ) ( )

[ ( ) ( )] ( ) ( )

( ) ( ) ( )

⩽

∑

∑

∑

αα

α α

α α

φφ= − ∂

= − +

= −

t

x

t

S A f f

S f f

xx

x x x x

x x x

d ,

d,

d

d,

, , ,

, , ,

0,

ii

i

i jij ij i j

i jij i j

,

,

(7)

the potential function φ is also a Lyapunov function of the deterministic counterpart dynamics. We further proved the dynamical equivalence between potential function and Lyapunov function [77], the correspondence between deter-ministic stable states and stochastic locally most probable states is achieved here. The decomposition has its application in engineering and physics [36, 76, 77]. Mathematically, our approach can be considered as using A-type stochastic inte-gration of the stochastic differential equation [76]. The A-type interpretation could be physically derived by taking zero mass limit of a 2n dimensional Newton’s equation with noise and the corresponding Klein–Kramers equation [76].

Applying traditional Ito or Stratonovich interpretation, on the other hand, we observe even for additive noise (constant diffusion matrix D) there can be inconsistency between sto-chastic and deterministic dynamics [78, 79], e.g. the deviation on the position of the stable fixed point as a function of the parameter in the non-detailed balance term in the first exam-ple of [78], leading to additional complexity in the modeling. The bridge between stochastic and deterministic dynamics is established within our framework: We can directly interpret stable states obtained from deterministic part differential equa-tions as the local maximum of the steady state distribution, the random trans itions between locally most probable states can be considered as transitions between corresponding stable states. Therefore, the potential landscape under A-type interpretation is robust with respect to noise. Notice that such robustness is also presumed to exist in many previous studies using tradition Ito or Stratonovich integration, e.g. calculating potential difference by path integral between given stable state and trans ition state obtained from deterministic equations without taking deviation of positions into account, resulting in complication and errone-ous outcome when noise strength is not small enough.

We omit the explicit dependency of noise in the following exposition within the framework of A-type stochastic integra-tion of the stochastic differential equations (1) using fi in (8):

=∑ ⋅+∑ ⋅

⋅+ ∑ ⋅

−x

t

a x

a x a xx

d

d 1

1

1.i u iu u

n

u iu un

v iv vn i

iu

iu iv (8)

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In discussing stochastic transitions, we may use diffusion matrix ( )D x derived from data [80], from standard chemical master equations, or simply a constant matrix at a first step. A conservation property of equation (8) is that all xi will remain in the interval [0,1]. Note that equation (8) allows the use of different cooperativity n and inverse dissociation a for each interaction between agents enabling a biologically motivated structural perturbation of the dynamics.

2.6. Perturbations on endogenous network

There are generally two types of perturbation on a dynami-cal system. The first type is structural perturbation that can reflect by alterations on the equations  describing the dynamics. There are several typical ways of structurally perturbing an endogenous network: (a) varying parameters in the differ ential equations, e.g. n and a in equation  (8); (b) blocking interactions between agents; (c) loss/gain of function of certain agents by mutation. The other type is the perturbation of the system’s state in the phase space with the equations left intact, we may refer it as positional per-turbation. Internal fluctuations and external inductions are two major sources. Accumulation of internal fluctuations may lead to a stochastic transition from one stable state to another through a transition state connecting them. External inductions caused by environmental change or drug treat-ment are able to ‘drag’ the system’s state from one attractive basin to another directly, leading to a state transition. These external perturbations can be impulsive, continuous, or has a specific pattern, reflecting the process of drug treatment. Potential landscape is a useful tool in understanding how all types of perturbations influence the system’s dynamics. It has stable states as valleys and transition states as passes and changes the height of the potential barrier in response to structural perturbations (see figure  1), within this con-text the risk factors causing stochastic transition between normal and cancer states can be discussed [3]. In addition, the basin of attraction for attractors can be visualized or calculated with the help of potential landscape as well. The mathematical framework mentioned in section 2.5 provides a method of constructing potential landscape for general stochastic dynamics [27, 76].

2.7. Cancer as robust intrinsic states

Based on previous arguments, we hypothesized that cancer is a state in the phase space of the endogenous network dynam-ics. The expression or activity pattern of agents of a cancer state should have clear biological significance that is consis-tent with experimental data. There are three additional condi-tions for the robustness of a cancer state:

(a) It is a dynamically stable state or an attractor of the deter-ministic differential equations. Within the framework of A-type stochastic integration, it is also the local max-imum of steady state distribution of the corresponding stochastic dynamics, the network will stay around it for considerably long time.

(b) It has a relatively large attractive basin, since obtained biological data, e.g. from patients of the same type of cancer, can have large inter-individual variance. The basin of attraction defines the phenotype regardless of the attractor type: fixed point, limit cycle or even chaotic, it includes all possible states of expression at molecular level.

(c) The cancer state should even be structurally stable, con-sistent with our modeling assumption at a core network level that are insensitive to the varying details of interac-tions between agents.

We show in the following section the existence of such a state fulfilling all the conditions discussed above in the endogenous network modeling of prostate cancer. Similar robust states are found for hepatocellular, gastric cancers, and acute promyelo-cytic leukemia as well [43–45]. Our modeling can be continu-ally refined with the accumulation of biological knowledge, which may show as new robust states emerging from the attractive basin of currently obtained ones.

3. Prostate cancer as a case study

Prostate cancer is one of the most frequently diagnosed can-cers and a leading cause of cancer deaths in men. It is one of the most intensively studied cancer with an abundance of related studies from clinical, pathological, and molecular aspects. With these available resources, prostate cancer was the first case that we tested the endogenous molecular–cellular network theory.

3.1. Core network construction

A mechanistic model of cancer at the whole genome scale is currently not feasible: There are many unknown players and most of the interaction details between players are also unavailable, e.g. in vivo parameters [23, 56, 57]. On the other hand, ‘master’ control genes or key pathways identified by previous studies on cancer as well as normal physiological functions [81–83] imply the existence of a hierarchical struc-ture in biological systems. Single-gene knockout experiments also demonstrate that most gene knockouts exhibit little or no effect on phenotype, while 19% of genes have been shown essential across a number of organisms [84]. Utilizing the inherent hierarchical structure in biological systems, a core network for cancer are obtainable under limited biological knowledge. The core network can be gradually extended or refined with the accumulation of knowledge, and ultimately reach a whole genome scale. We demonstrate in the following the brief construction of the core network for prostate cancer.

3.1.1. Cell cycle. The commitment stage of the cell cycle is represented by the phosphorylation of the Retinoblas-toma protein pRb [82]. When entering the cell cycle, genes encoding D-type cyclins (Cyclin D) are induced in response to mitogenic signals [85]. Cyclin D assembles with catalytic partners, Cdk4 and Cdk6, as the cell progresses through the G1 phase. There are two immediate positive feedbacks on pRb

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phosphorylation, by Cyclin D/Cdk4,6 complex and by Cyclin E/Cdk2 complex. The effect of pRb phosphorylation is addi-tive. Both Cyclin D/Cdk4,6 and Cyclin E/Cdk2 need to be acti-vated to fully phosphorylate pRb. Cyclin D dependent kinases initiate pRb phosphorylation in the mid G1 phase after which Cyclin E/Cdk2 becomes active and completes this process by phosphorylating pRb on additional sites. Phosphorylation of pRb protein releases E2F proteins to allow transcription of their target genes, which include those required for DNA rep-lication, such as E2F itself and Cyclin E. Myc [65, 86] is a positive regulator of cell cycle progress. Its activation depends on mitogenic activity. Suppression by pRb family member p107 might be one of the regulatory factors. Activated Myc has a wide range of effects on the cell cycle through stimu-lating the activity of Cyclin E/Cdk2 complex and Cyclin D/Cdk4,6 complex, suppressing p27 function, down regulating p21 levels, and upregulating E2F expression [86]. Myc also influences apoptotic activities by blocking p21 induction as well as through p53 and upregulating p53 [65]. In quiescent cells, the levels of p27 are relatively high while p21 levels are low, but usually increase during G1 phase progression, while p53 is a major activator of p21 transcription. Among the Cdk inhibitors [87], only p21 and p27 are explicitly included for their interactions with other proteins in the model.

3.1.2. Apoptosis. Apoptosis is carried out by effectors (cas-pases 3, 6, and 7) [88]. Among them, caspase 3 is the major one, here it is used to represent the caspase cascade (path-way). The assumption is that the activation of Caspase 3 is enough to activate other effectors. Three proteins are neces-sary and sufficient to activate Caspase 3 in vitro: Cytochrome c, Apaf-1, and Caspase 9. Among the inhibitor of apoptosis (IAP) family of proteins, the ubiquitously expressed, X-linked inhibitor of apoptosis (XIAP) is included in the model. XIAP is an endogenous inhibitor of cell death that functions by suppressing caspases 3, 7, and 9. XIAP [89] inhibits Apaf-1, which mediates activation of procaspase-9 and activity of processed Caspase 9. In the model Cytochrome c, Apaf-1, and Caspase 9 are treated as a unit (represented by Cytochrome c). Therefore the whole Caspase pathway triggered by Caspase 9 to mediate chemical-induced apoptosis is included in effect. XIAP is also a part of the positive feedback loop in apoptosis activation through cleavage and inactivation by caspases [89]. Another positive feedback to apoptosis is Smac released from mitochondria during apoptosis that may neutralize IAP func-tion [89]. XIAP is upregulated by Ras through Akt activation [90]. Plasma membrane-associated ‘death receptors’, such as Fas receptors TNFR1 and TNFR2, recruit procaspase 8, and result in aggregation of Caspase 8 molecules, triggering their autoactivation. Activated Caspase 8 can then directly process Caspase 3 leading to apoptosis [91].

The Bcl-2 family of proteins regulates Cytochrome c released from mitochondria as well as caspase activation [92, 93]. Some of the proapoptotic proteins have multiple Bcl-2 homology domains, BH1-3 proteins, such as Bax and Bak; some have only one, such as BH-3 only proteins, including Bid, Bim, Bad. The BH1-3 proteins may be the primary effec-tors of cell death, and the BH-3 only proteins may generally

act in concert with Bax or Bak. For example, Bid activates Bax to produce membrane opening. This process requires cardi-olipin that is inhibited by Bcl-xL. Anti-apoptotic protein Bcl-2 captures activated Bax [94]. Bcl-xL proteins also block apop-tosis by inhibiting Cytochrome c release [95]. Apoptosis and cell cycle regulation are also coupled through Myc-induced apoptosis by engaging Bax [96].

3.1.3. Growth factors. Both cell cycle and apoptosis are reg-ulated by growth factors and cytokines. Insulin like growth factor-1 (IGF-1) [97] stimulates cell proliferation and inhibits apoptosis. The activated IGF-1 receptor signals through the MAP kinase (MAPK). IGF inhibition of apoptosis is associ-ated with Bcl-xL upregulation [98]. p53 regulates effect of insulin like growth factor-I by downregulating its receptor lev-els [99]. Prostanoids can induce angiogenesis by stimulating vascular endothelial growth factor (VEGF) [100]. Integrins act as both cofactors and directly interact with growth fac-tors such as VEGF and EGF [101, 102]. There is crosstalk between interleukin 6 and VEGF [103].

3.1.4. Metabolism and stress response. Growth factors acti-vate Akt by engaging the PI3-kinase/Akt pathway [104]. The Akt activating ability of PI3-kinase is suppressed by lipid phosphatase PTEN [105]. In addition to the PI3 kinase/Akt pathway, the mitogen-activated kinase (MAPK) pathway is a critical pathway for proliferation that PTEN regulates. PTEN also inhibits the cell cycle by upregulating p27 [106]. PTEN expression is upregulated by p53 [107] at least partly through p53 regulation of PTEN stability [108]. NF-κB downregulates PTEN expression [109]. Ras targets multiple MAP kinase cas-cades [110]. In addition, Ras downregulates Fas expression [111], while TNF-α sensitizes Fas [112]. Akt pathway acti-vation has diverse effects on the cell [113, 114]. It mediates metabolic changes, upregulates GLUT1 and glycolytic gene expression, decreases the cell’s sensitivity to apoptosis and increases the cell’s growth rate. The activation of hypoxia-inducible factors (HIF) can be mediated by high glycolytic rate [115].

3.1.5. Immune response. Prolonged activation of the NF-κB pathway in the prostate cells eventually has detrimental effects on these cells. The transcription factor NF-κB plays a cen-tral role in the regulation of many immune and inflammatory responses [116]. NF-κB can be activated by signals associ-ated with stress and injury including cytokines such as tumor necrosis factors (TNFs) and oxygen free radicals. Activating NF-κB requires phosphorylation of its inhibitory proteins iκB. Such activation is usually transient, at least partly because the NF-κB inhibitor iκB is among downstream targets of NF-κB. NF-κB plays an important role in the expression of a large number of inducible genes involved in stress and inflamma-tory responses including interleukin IL-6. NF-κB activation leads to transactivation of MAPK pathway and Akt activations of NF-κB. NF-κB upregulates cyclooxygenase-2 (COX-2) expression [116]. Overexpression of COX-2 enzymes leads to increased amounts of prostanoids in tumors, which affects a wide range of downstream targets. Upregulation of COX-2 is

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also mediated by MAPK [100]. HIF-1 can be upregulated by growth factor through PI3-kinase/Akt pathway [117]. HIF-1 activation upregulates epidermal growth factor (EGF) recep-tor pathway [118]. Integrins induce a wide range of intra-cellular signaling events including the activation of Ras and MAP kinase [119]. Integrin signaling preserves cell viability by increase Bcl-2 expression. To include the feedback effect, immune cells such as macrophage and T cells recruited into cancer growth are included in the model [120]. IL-6 and tumor necrosis factor alpha (TNF-α) are mainly secreted by immune cells. IL-10 is secreted by the immune cells directly

stimulated by TNF-α to regulate immune response. Such activating inhibition is abundant in the endogenous network. Another such example is MAP kinase phosphatase that keeps a cap for MAPK activation [121].

3.1.6. Extracellular matrix. E-Cadherin as well as integrins are important in cancer invasion [122]. TNF-α may play a role in E-Cadherin expression [123]. E-Cadherin suppresses tumor growth and is mediated by elevated expression of p27 [124]. The cytokine transforming growth factor beta (TGF-β) modulates cell growth as well [125]. In addition to its

Figure 6. Stable states and transition states obtained from differential equations and comparison with results from Boolean dynamics. Each stable/transition state is specified by a column vector of the expression or activity levels of agents in the endogenous network for prostate cancer. All values are normalized between [0,1] denoted by gradually varying colors. The comparison shows a clear consistency between results from differential equations and Boolean dynamics. And transition states are additional information provided by differential equations. The stable states with name D for differentiation, P for proliferating, A1 to A5 for apoptosis, C for cancer, and I for inflammation signatures. Transition state with name D-P means it connects state D and P, the system initially at D-P transition state under a small perturbation will go to either D or P. Here GR, PS, and ECM denote growth factors, prostate specific factors, and extracellular matrix modules separately.

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anti-proliferation effect, TGF-β also inhibits E-Cadherin expression. Latent TGF-β may be activated by integrins that in turn respond to inflammation [126].

3.1.7. Prostate specific factors. In the hierarchical modeling of prostate cancer [42], the prostate specific module is char-acterized by the prostate specific sub-network with 60 nodes. Here in the core network, androgen receptors serve as a repre-sentative. Androgen receptor is a transcription factor targeting a variety of genes in prostate cells when activated. Androgen receptor can interact with growth factors and cytokines [127]. In the serum, IGF-I is bound predominantly to IGFBP-3, which inhibits IGF-I activity. The androgen receptor target gene, prostate-specific antigen, is capable of proteolytic cleav-age of IGFBP-3, resulting in the release of IGF-I [128].

3.2. Modeling results

The core endogenous molecular–cellular network constructed for prostate cancer is shown in figure  3. It has 38 agents from 7 modules, with 115 interactions, 71 activating and 44

inhibiting. We converted the network into Boolean dynamics and differential equations  following the rules demonstrated in figure  5. The first task is to identify robust states in the phase space. The computational complexity for the exhaustive search of attractors of a Boolean dynamics increases expo-nentially. When n is large, a complete search is not feasible. Stable states of differential equations may be computationally more difficult to find. The robustness of the cancer state as well as other normal functional states, however, provides us a way to solve this problem. As discussed in section 2.7, cancer state has a relatively large basin of attraction, due to the inter-individual variance observed from data. The task reduces to find out stable states with non-small attractive basins, which can be done by simulations with a large number of random ini-tial conditions. We did simulations for Boolean dynamics and differential equations stopping by not finding any new stable state for three orders of magnitude in number of random ini-tial conditions. Both dynamics generate 10 stable states with non-small basin of attraction, see figure  6. In addition, we used Newton’s iteration method obtaining fixed points from the differ ential equations, also running with plenty number

(a)

(b)

Figure 7. Calculated potential landscape. (a) Landscape on an invariant surface in the phase space delineates multiple paths for stochastic transitions. (b) Projected landscape on the surface determined by three stable states.

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of initial points, and the three orders of magnitude stopping condition if no new fixed point is found. We exactly obtain the 10 stable fixed points (stable states), and further acquiring 16 transition states with one unstable eigenvalue (positive real part) of its Jacobian matrix, together with 13 hyper-transition states with more than one unstable eigenvalues. Five trans-ition states connecting the first five stable states are provided in figure 8.

Potential landscapes can be calculated directly using the method proposed in our previous works [27, 42] as shown in figure 7. The projection graph of the potential landscape on the surface determined by three high-dimensional stable states is given in figure 7(b). If the dynamical system has a structure, such as the attractive invariant surface (shown in figure 7(a)) found in the present prostate network [42], a direct visualiza-tion of the potential landscape on the invariant surface can be realized. This invariant surface has stable states D, P, C, I at the corners. Starting within the invariant surface, a trajectory of the differential equations will remain there. Nearby trajectories are also attracted by the surface, leading to a higher potential value in the neighborhood phase space. Thus this figure demonstrates the most probable routes for stochastic transition between the biologically meaningful states D, P, C, and I, a key structure of the network dynamics. We can further demonstrate even with-out a specific structure, a sketch of the landscape is obtainable for general systems. By small random perturbations for the dynamical system initially at a transition or hyper-transition state, we can follow the trajectories of differential equations to see which state the system will finally reach. Thus a connec-tion graph for transition and hyper-transition states is obtained, see figure 8. It shows us the possible stochastic transition paths from one robust state to another and an overview of the global dynamical structure in the phase space.

3.3. Robustness of the state corresponding to prostate cancer

We examine in this section whether our modeling results sat-isfy the robustness conditions proposed in section 2.7. Firstly, the expression or activity pattern of the agents of the cancer state and normal functional states identified is compared with the clinical knowledge in figure 9. The comparison shows a clear correspondence and manifests the biological signifi-cance of the obtained stable states. More detailed comparison with data can be seen in [42]. For the robustness conditions:

(a) All states are dynamically stable, this condition is auto-matically satisfied.

(b) The stable states are obtained by simulating trajectories of the differential equations with a large amount of uniformly distributed random initial conditions. For a run with one million initial conditions as an illustration, the number falling into each state proportional to the size of the attrac-tive basin would be, D: 10048, P: 65814, A1: 3508, C: 281502, I: 193365, A2: 2400, A3: 9100, A4: 153277, A5: 125477, A6: 155509. The cancer state C has a relatively large basin of attraction consistent with the inter-individual variance observed from data of cancer patients.

(c) The structural stability is shown by random parameter tests in figure  10. By using random real number Hill coefficient nij in the range [1,30] that enables strong to no cooperativity for each of the 115 interactions = …i j, 1, , 38, we observed that more than 98% cases

in 50000 tests, all the 10 stable states are re-obtained. Different forms of differential equations  are tested as well, without influencing these stable states. The result further implies that our modeling is not sensitive to the varying interaction details between agents. Note that the robustness demonstrated here is based on the ‘collective-ness’ topology of the underlying bionetwork (random network generates very few stable states) as well as real-izing equations  (consistent to the results from Boolean dynamics).

Therefore, we may conclude that a robust cancer state and normal physiological states are generated by our endogenous molecular–cellular network modeling.

Note that the endogenous molecular–cellular network is constructed under normal physiological conditions. Both nor-mal and cancer robust states are generated by the same net-work. From such a viewpoint, pro-oncogene, oncogene, tumor suppressor gene may all have their roles in normal functions. Conversely, genes or pathways responsible for developmen-tal processes and physiological functions may also underpin the cancer state. In addition, the interactions between these genes and pathways are regulated by feedback loops and inte-grated as a nonlinear dynamical system, instead of linear addi-tive notions. Therefore, the prevailing nomenclature may be misleading.

Figure 8. Stable states connected by transition/hyper-transition states. The figure sketches the global structure of the network dynamics. D, P, A1, C, and I are five functional robust states with expression or activity values listed in figure 6. The brown nodes are transition states and red ones are hyper-transition states with more than one unstable eigenvalues for its Jacobian matrix.

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3.4. Modeling hepatocellular and gastric cancers as well as acute promyelocytic leukemia

In another three of our recent works [43–45], we have con-structed endogenous network models for hepatocellular and gastric cancers as well as acute promyelocytic leukemia, with explorations on different facets of the framework supplemen-tary to the material on prostate cancer. By comparing the three network models for different cancers, we find that they share a same core network and are distinguished by different tissue specific modules. The observation suggests that a unique and unified endogenous network may exist.

3.4.1. Hepatocellular cancer. The model for hepatocel-lular carcinoma [43] consists of 7 modules, 42 agents, and 123 interactions. In the paper [43], cancer as robust state of the endogenous network dynamics is further demonstrated by exploring the network wiring: The cancer state is main-tained by a set of activated agents constituting a sub-net-work with positive feedback loops shown in the left panel of figure 11. Such wiring structure determines that a cancer state is stable under perturbation. The normal physiological states are maintained by different sub-networks that mutually inhibit with those of the cancer states, shown in the right panel of figure  11. The result clearly elucidates that a single endog-enous network is responsible for the multiple robust pheno-typical sub-networks.

3.4.2. Gastric cancer. The paper for gastric cancer [44] con-tains a network with 8 modules, 48 agents, 215 interactions and focuses on the discussion of cancer heterogeneity. The heterogeneity in carcinogenesis may be a result of multiple possible transition paths shown in figure 12 and in figure 7(a), as well as the existence of multiple cancer states. A cancer state has a large basin of attraction that allows various expres-sion or activity levels of molecular–cellular agents, which may be another contribution to the cancer heterogeneity.

3.4.3. Acute promyelocytic leukemia. Acute promyelocytic leukemia is considered as one of the ‘simplest’ cancers, in the sense that it generally has a unique and uniform phenotype, well defined mutations, and can be ‘cured’ by a simple drug combination of all- trans retinoic acid (ATRA) and arsenic trioxide (ATO) for above 90% cases. We have also constructed an endogenous network model for APL [45], which contains 8 modules, 81 nodes, and 416 interactions collected from 357

references. Our computer simulations of induced transitions identify molecular targets for the drug combination. In figure 13, we show both the effective network of attractors corre sponding to APL and normal progenitor states and discovered molecu-lar targets for induced switching. In addition, the modeling shows APL and normal states mutually suppress each other, both in ‘wiring’ and in dynamical cooperation.

4. Discussion

4.1. Systems, quantitative, and mechanistic view of cancer

We have obtained dynamically robust cancer state and normal functional states in our modeling, and explained carcinogen-esis as a state transition on the potential landscape. Evolution may play a crucial role in the origination of cancer, since the

Figure 9. Consistency at the modular level: Expression or activity pattern of agents for cancer state and functional states versus clinical knowledge. For more detailed comparisons, please see [42].

Figure 10. Random Hill coefficients test demonstrates modeling robustness. In this test, we used the equations listed above that allow different Hill coefficient nij for each of the 115 interactions i j, 1, , 38= … , where agent j to agent i has an interaction. With uniformly distributed real random numbers n 1, 30ij [ ]∈ and a 2ij

nij= , 50,000 tests showed that all 10 stable states have more than 98% recurrence rate. Since each x is normalized between 0 and 1, the setting of a 2ij

nij= enables a regular condition a x 1ijnij⋅ =

thus a x1 1 0.5ijnij/( )+ ⋅ = for x = 0.5. Note that the u∑ and the v∑

mean summation over all agents activating and inhibiting agent i separately. The result exhibits that our modeling may not be sensitive to the varying interaction details.

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endogenous network responsible for cancer is evolutionarily formed. Cancer state may be previously functional in the history of evolution but no longer optimized for the current interest of the whole organism. Similar opinion on cancer as atavism [129] was proposed. The view of cancer as a robust state integrating interplay among multiple players separates cancer from a solely genome determined disease. Genomic mutation is among these players but not the only one. In the traditional view, since cancer cells that have accumulated a series of key mutations building up cancer hallmarks are uncontrollable, this seems to be the only way for cancer ther-apy to kill/remove all such cells. Our view, however, provides a different angle: stochastic transitions are bidirectional, since there are spontaneous transitions to a cancer state, the trans-ition from the cancer state to a normal state is also possible, which naturally explains the spontaneous regression of can-cer [11]. Differentiation therapy for acute promyelocytic leu-kemia provides another paradigm [130]—by drug treatment combining all- trans retinoic acid and arsenic trioxide, the leu-kemia cells are induced to normal-looking differentiated cells. Nevertheless, the prevailing view on APL is that it is a genetic disease caused by a fusion between the retinoic acid receptor α (RARα) and promyelocytic leukemia (PML) gene through chromosomal translocation. Evidence like RARα-PML fusion is detectable in patients under long-term remission [10] and is usually overlooked. Recent experiment also shows that the ‘reprogrammed’ B-cell acute lymphoblastic leukemia (B-ALL) cells with another key carcinogenic BCR-ABL1 translocation appeared to lose their carcinogenicity [131]. Cancer patients without mutations [8, 9] are also reported. These all support our view of cancer as a robust state.

4.2. Stable, metastable, and critical states

With the three endogenous network modeling [42–44] reviewed, we further discuss a few qualitative and natural predictions from the endogenous network theory—two very

different groups of outcomes are implied in figure 1: One is nearly deterministic and the other is of statistical mechanics.

Near the deterministic limit, the noise strength is small, �ε 1, the potential function φ (equation (4)) for the landscape

of figure 1 may adequately describe three typical situations for the normal functional state: stable, metastable, critical or unsta-ble. They may correspond to the three physiological states: health, sub-health, and disease (cancer). Such progress may be represented by the variation of system parameters α, which can be both genetic and epigenetic. Two specific predictions can be made here: 1. Because of multi-stability there should be a hysteresis in varying the system parameters α back and forward from normal and caner states. From the medical point of view, this implies that cause(s) to cancer are typically not the cure(s) to cancer. 2. Near the critical point, at least two states, one stable state and one transition state, are merged. Hence one should expect a dramatic increase in fluctuations, which can be good indication of getting into cancer state. Both features are clearly observed in endogenous network modeling [43].

If the noise is large, its contribution can be considered in the form of entropy, encoded by the free energy F in statistical mechanics [132]:

∫φ

αα

= − −⎡⎣⎢

⎤⎦⎥ε ε

εF x

x, ln d exp

,( ) ( ) (9)

Here all possibilities are considered by integration over the whole phase space. By varying the noise strength, say, from small to large, there may be three typical situations again corre sponding to what described in figure 1: one single phase in thermodynamic sense, mixed phases and when passing through a critical point the system moves into another totally different phase. The latter two of such scenarios are similar to the first order and the continuous phase transitions in ther-modynamics, also noticed by other researchers [37]. It may display what we found in the small noise case as well: hyster-esis for the first order phase transition and large fluctuations at the critical point. Its implication for cancer progression is

Figure 11. Cancer as a robust state demonstrated by network wiring. The normal and cancer states are obtained for the endogenous molecular cellular network modeling of hepatocellular carcinoma. Two mutually inhibited groups of agents with positive feedback inside are identified as responsible for sustaining normal and cancer robust states. The left panel shows that a single network can generate multiple subnetworks correspond to distinct phenotypes. The right panel delineates that positive feedback loops maintain robust states and controlling other modules as well as downstream pathways. Reproduced from [43] with permission of The Royal Society of Chemisty.

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very different from that of small noise limit. The treatment of cancer is very different too: It is unlikely that such exten-sive integration over phase space would be caused by a single molecular or epigenetic agent. Rather, it may be a result of large scale influence, such as immune and/or stress responses. This important scenario has not been adequately explored yet.

4.3. Robust states as ‘phase space’; further abstraction

The endogenous network theory shows the multiple robust states in its potential landscape (figures 1, 7, 8), which can be quantitatively specified individually (figures 6, 9; [42–45]). The experimental results such as those of Phage lambda genetic switch ([49]) and of stem cells ([133]) suggest this possibility, too. All states should be there even for the tis-sue in the normal state. This feature implies a higher level of abstraction on the modeling: One can directly use such robust states as the ‘phase space’ and set up a dynamical equation to describe the Markov process

( ) ( )∑=P t

tW P t

d

dl

mlm m (10)

where Pl is the probability for the lth robust state at time t and Wlm is the transition rate from the mth to the lth state. The trans ition rate matrix W satisfies the condition ∑ =W 0l lm such that the total probability ( )∑ P tl l is a constant. Evidently, W can be computed from equation (1) or (4). In directly apply-ing equation (10) to understand observations, it can be showed [134] that equation (10) has the same structure as that of equa-tion (1): the potential function, the anti-symmetric matrix and the corresponding diffusion matrix. In this type of modeling of a tissue, its normal, cancer, or others states, are described by the probability distribution function {Pl}.

4.4. Clinical and pharmaceutical application: searching of biomarkers and drug targets

Inspired by the clinical success of the differentiation therapy for APL as well as laboratory approaches such as reprogramming of B-ALL cells, we discuss in this section the possible clinical

or pharmaceutical application of endogenous molecular–cellular network modeling. The robustness of cancer has been demon-strated in its response to cancer therapies. In general, no effec-tive drug treatment has been found for most cancers. The ATRA and ATO treatment for APL is not as effective for other myeloid leukemias. The development of drug resist ance makes the situ-ation even worse. Since the screening of drug combinations is unrealistic with exponentially increasing working load, there is eagerness to develop a computational method integrating existed biological knowledge to reduce the huge space of possibility. The endogenous network approach may be a candidate. A drug treatment of cancer may be considered as an induced positional perturbation of the cancer state. If such a perturbation success-fully drag the system’s state out of the attractive basin of can-cer, the drug combination may be effective. The development of drug resistance may be understood as the diverse responses to treatment of cells in the multiple cancer states: Some cells are dragged to normal while others that are stable under perturba-tions from the treatment are left cancerous. The endogenous net-work approach directly provides a platform for in silico search of key agents to induce that will trigger a state transition. Drug resistance might be solved by using different drug combinations in series. We have exhaustively searched all possible 4 or less combinations of agents for escaping the cancer state shown in figure 14. These key agents found may serve as targets for the guide of using proper drug combinations. In addition, biomark-ers may be recognized as differentially expressed agents or their downstream genes (figure 6). A possible way of personalized medicine can be proposed: Through measuring the expression or activity levels of the agents or their downstream genes for a patient as an initial condition of the differential equations, simulating the trajectory and see where it goes. In a model with multiple stable states of different prognosis, we can generate a personalized advice for treating this particular patient.

4.5. Comparison with other network modeling

The endogenous network modeling of cancer is aimed to reveal the core regulatory mechanism of the genesis and development of cancer. There are plenty of high-throughput data based statistical

Figure 12. Multiple transition routes for the development of gastric cancer and corresponding dynamics for the activity or expression level of key molecular–cellular agents. A1 is a cell cycle off state with gastric differentiation signature. P1 and P2 are possible cancer states. Four paths through transition states from A1 to P1 and P2 are displayed. Reproduced from [44]. CC BY 3.0.

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approaches [135–138] that have been proposed with a similar goal. Because of the complexity of cancer and noise in the data, a satisfactory modeling has not been established. Nevertheless, results from these studies may be used as validation of endog-enous network modeling. Approaches [135] utilizing both accu-mulated low-throughput knowledge and high-throughput data are also inspiring for our current method. Besides, Boolean dynamics is used to analyze the network dynamics of cancer [72] and other biological systems [139]. Transition states and hyper-transition states are not available in their modeling results. As we have demonstrated, these points play a critical role in carci-nogenesis and transitions among biologically functional states.

Thus a comprehensive modeling may include stochastic differ-ential equations under A-type integration. In addition, it has been argued that a core network to characterize the most basic features of a cell’s behavior may require more than 20 nodes [3], crucial robust states are absent in smaller network modeling [139, 140].

5. Looking forwards

The endogenous molecular–cellular network theory offers a starting point for the mechanistic study of cancer from the systems biology perspective [141], integrating existing bio-logical knowledge through stochastic and nonlinear network

Figure 13. Effective network of attractors corresponding to normal progenitor and APL states and induced transition between them. (a) Effective sub-networks for normal progenitor and APL attractors, S4 and S1 in figure 3 of [45]. The effective sub-networks are drawn by selecting active nodes (value > 0.5) in S1 and S4 as well as molecular interactions among them. Within orange/blue dotted frames are the active nodes in S4/S1. Solid green/red lines denote activation/inhibition and double arrowed green/red lines represent mutual activation/inhibition. These two attractors have active nodes in common and distinct ones. Inhibitory interactions dominate the differential ones of the two attractors such as those among BMP, Runx1, Stat5, and SHH. By an exhaustive search of all three-node combinations with total number 682 560, we find a few combinations of targets for inducing transitions between S1 and S4. (b) Induced transition from APL attractor S1 to normal progenitor attractor S4. Starting from APL attractor S1, at time t = t0, three nodes are changed to: SHH = 0, Runx1 = 1, BMP = 1. Trajectory of the time course shows that after such an induction, the state of the system will switch from S1 to S4. The trajectory of selected nodes during the transition process is shown there. Reproduced from [45]. CC BY 3.0.

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dynamics in the phase space. We have modeled four different types of cancer with accumulated knowledge from individual biochemistry and molecular biology experiments. The model-ing results are verified by microarray profiling of tumors. At the modular level, the consistency between theory and experi-ment is 100 percent for all models. At the molecular agent level the consistent is typically around 70% or more, near the experimental error limit (figure 2). Such a consequence may suggest that the endogenous network theory is on the right track. Nevertheless, numerous challenges from experimental, theoretical, and computational aspects are still lying ahead. We list below a few major problems from current standpoint:

(i) A foremost question is that to what extend an evo-lutionarily conserved endogenous molecular–cellular network can be uniquely determined and systemati-cally verified similar to that of metabolic network. We have constructed core network models for prostate, hepatocellular, and gastric cancers as well as acute pro-myelocytic leukemia, verified by microarray profiling of tumors and other experiments. We find that these networks overlap strongly. Thus, the answer to this question appears to be positive. In addition, since an individual is developed from one fertilized egg, which has the genome shared by all others in later stages, one endogenous network is implied.

(ii) The endogenous network theory generates distinct pre-dictions qualitatively different from those of somatic mutation theory, such as cancer-like cells (or neoplasm) without genetic defect can arise and the existence of spontaneous remission/regression of cancer patients without treatment. Despite occasionally reported cases [8, 10, 11], a systematic documentation and study of

related experimental and clinical observations would be necessary.

(iii) The framework for decomposing a general stochastic dynamics to obtain a potential landscape [27] has been found useful in applications such as in the quantitative modeling of the Phage lambda genetic switch [49]. Potential function has been explicitly constructed for general fixed points [142], limit cycles [58], as well as chaotic attractors [143]. Nevertheless, a rigorous math-ematical exposition of the framework is still in demand.

(iv) We have been able to compute multiple normal cell types. A natural question would be: can the endogenous molecular–cellular network model generate all normal cell types emerged in the developmental processes? Even for limited but well known cases, such as hemat-opoietic differentiation [144], an explicit demonstration will be extremely valuable.

(v) Incorporating spatial effects of cell populations in tis-sues [145] is a major direction to extend endogenous network modeling: different cell types may be of the different states of the endogenous network, and, the interaction among them may certainly induce their mutual transitions and other effects.

(vi) The endogenous network should be used to predict the influence of certain mutations to the stability of the robust states. This ability has already been dem-onstrated in one of the simplest endogenous networks [49, 146] as well as in predicting gain or loss of func-tions for genetic mutations in hepatocellular carcinoma [147]. With this ability, the effect of mutations can be incorporated as a real prediction of a network. The challenging task is to quantify such influence for indi-vidual patients based on their genomic profiles in order to produce better prognosis.

(vii) Most of the effort so far has been focused on the land-scape and associated phenomena. There is no substantial exploration of other dynamical consequences directly associated with quantities such as the friction matrix S and the transverse matrix A in equation (4). For example, it is known that the transverse matrix may generate a finite stationary current, and, dynamically it makes the trajectory not following the steepest descent path in the potential landscape [27]. One may even ask whether or not an analogy to (linear) response theory in physics can be established here and wonder what new insights such endeavor can generate on the cancer genesis and pro-gression. Another natural question arises, too: whether or not more specific knowledge on interactions between agents such as in vivo kinetic parameters can be restored back into those normalized quantities in equation (8), to enable better contact with experimental data, even though one may not expect as detail and realistic comparable to that of Phage lambda genetic switch [49, 51].

(viii) Based on evidence from evolution [53], a comprehensive and evolutionarily conserved endogenous network may contain several hundreds of nodes. First, current core networks should be extended to more comprehensive networks of about 200 nodes. It is likely that more wet

Figure 14. In silico search on induced transitions reveals possible drug targets. The name of an agent in red represents up-regulation and in green for down-regulation. The names enclosed in brackets denote the induction of these agents is simultaneous. The names separated by slash mean multiple choices.

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biological studies are needed to find out such compre-hensive networks. In addition, to analyze the network dynamics at such a scale, better theoretical and compu-tational tools are needed.

(ix) How do we expand the constructed models towards a whole genome scale and the whole cell/tissue? For example, metabolism has been important and meta-bolic network dynamics has been developing [57]. Our cur rently constructed networks are of minimal core net-work with dozens of nodes, more functional modules involving newly discovered players should be included.

The accumulation of data and knowledge on biological path-ways and their interactions will enable many other types of cancer and complex diseases such as osteoporosis [148] to be studied within the framework of endogenous molecular–cellular network. The ultimate goal would be the construc-tion of a unique genome scale endogenous molecular–cellular network model that can generate many robust states including known cell types, functional states, as well as diseases; is the unification theory. In pursuing such theory, evolution clearly plays a dominant role: first as historical fact and constraint to find out the endogenous network and then as the dynami-cal framework for analysis and prediction. If such a goal is achieved, dry experiments, e.g. the searching of biomarkers and drug targets or testing drug effects in silicon will be on an equal footing with wet biology at the laboratory bench, and provide insights in hospitals on diagnosis, identifying risk fac-tors, treatments, and prognosis. This research program may offer an opportunity for physicists to prove that they can make fundamental contributions in attacking biological complexity, not merely excellent tool makers. As Max Delbrück, Francis Crick and others had already demonstrated [149], there are general rules/laws at biological and medical levels, too.

Acknowledgments

Numerous wet biologists and physicians have been helping us in the development of the cancer theory reviewed here. Some even made important contributions at various stages. In par ticular, we should thank David Galas, Jianren Gu, Leroy Hood, Bingya Liu, Jerry Radich for their stimulating discus-sions and involvement. This work was supported in part by the Natural Science Foundation of China No. NSFC91329301 and No. NSFC9152930016; and by the grants from the State Key Laboratory of Oncogenes and Related Genes (No. 90-10-11).

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