HMGB1 signaling pathway

Our HMGB1 signaling pathway model is illustrated in Fig. 1. It includes 31 molecular species (6 tumor suppressor proteins), 59 chemical reactions, and three different signaling pathways activated by HMGB1: the RAS-ERK, Rb-E2F and p53-MDM2 pathways. Since the interaction between HMGB1 and its receptors TLR and RAGE is not clear at the mechanistic level, RAGE is used to represent all the receptors in our model in order to reduce the number of unknown parameters. We now briefly discuss the three pathways and their crosstalk. We denote activation (or promotion) by →, while inhibition (or repression) is denoted by ⊣.

The p53-MDM2 pathway is regulated by a negative feedback loop [26]: PI3K → PIP3 → AKT → MDM2 ⊣ p53 → MDM2, and a positive feedback loop: p53 → PTEN ⊣ PIP3 → AKT → MDM2 ⊣ p53. The protein PI3K is activated by the toll-like receptors (TLR2/4) within several minutes after TLR2/4 activation by HMGB1 [27]. In turn, PI3K phosphorylates the phosphatidylinositol 4,5-bisphosphate (PIP2) to phosphatidylinositol (3,4,5)-trisphosphate (PIP3), leading to phosphorylation of AKT. The unphosphorylated oncoprotein MDM2, which is one of p53's transcription targets [28], resides in the cytoplasm, and cannot enter the nucleus until it is phosphorylated by activated AKT. The phosphorylated MDM2 translocates into the nucleus to bind with p53, inhibiting p53's transcription activity and initializing p53 polyubiquitination [29], which targets it for degradation. Also, p53 can regulate the transcription of PTEN [30], a tumor suppressor protein, which can hydrolyze PIP3 to PIP2, thereby inhibiting the activation of AKT and MDM2.

The RAS-ERK pathway is the activation sequence: RAS → RAF → MEK → ERK → CyclinD. Activation of RAGE by HMGB1 leads to RAS activation, which in turn activates its effector protein RAF. Activated RAF will phosphorylate the MEK proteins (mitogen-activated protein kinase kinases (MAPKK)), leading to the phosphorylation of ERK1/2 (also called MAPKs). Activated ERK can phosphorylate some transcription factors which activate the expression of the regulatory protein CyclinD and Myc, enabling progression of the cell cycle through the G1 phase. K-RAS, a member of the RAS protein family, is found to be mutated in over 90% of pancreatic cancers [31].

The Rb-E2F pathway is composed of the interactions: CyclinD ⊣ Rb ⊣ E2F → CyclinE ⊣ Rb. The Rb-E2F pathway regulates the G1-S phase transition in the cell cycle during cell proliferation. E2F is a transcription factor that can activate the transcription of many proteins involved in DNA replication and cell-cycle progression [32]. In quiescent cells, E2F is bound by unphosphorylated Rb - a tumor suppressor protein - forming an Rb-E2F complex which inhibits E2F's transcription activity. E2F will be activated and released when its inhibitor Rb is phosphorylated by some oncoproteins (CyclinD and Myc in Fig. 1), leading to the transcription of CyclinE and Cyclin-dependent protein kinase 2 (CDK2) which promote cell-cycle progression. CyclinE, in turn, continues to inhibit the activity of Rb, leading to a positive feedback loop [33–35]. Fig. 1 shows that the activity of CyclinD-CDK4/6 (only CyclinD is shown in Fig. 1) is inhibited by the tumor suppressor protein INK4A, which is inactivated in up to 90% pancreatic cancers [36].

The crosstalk between these pathways can influence the cell's fate since the three signaling pathways in HMGB1 signal transduction are not independent. As shown in Fig. 1, the oncoprotein RAS can also activate the PI3K-AKT signaling pathway; the tumor suppressor ARF protein induced by E2F can bind to MDM2 to promote its rapid degradation and stabilize p53. Furthermore, it has been experimentally observed [13] that the p53-dependent tumor suppressor proteins P21 and FBXW7 can inhibit the activity of cyclin dependent kinases (In Fig. 1, we use P21 to represent both P21 and FBXW7's contribution). Mutations of RAS, ARF, P21 and FBXW7 have been found in many cancers [31, 36, 37]. One of our aims is to investigate how these mutations might influence the cell's fate.

In the HMGB1 model, all substrates are expressed in the number of molecules; proteins with the subscript "a" or "p" correspond respectively to active or phosphorylated forms of the proteins. For example,

We denote the mRNA transcript of MDM2 by mdm 2. We assume that the total number of active and inactive forms of the RAGE, PI3K, PIP, AKT, RAS, RAF, MEK, and ERK molecules is constant. For example, AKT + AKT _p = AKT _tot , PIP2 + PIP3 = PIP _tot . We sometimes use CD to stand for the CyclinD-CDK4/6 complex, CE for CyclinE, and RE for the Rb-E2F complex.

The p53-MDM2 and RAS-ERK pathways have been studied individually using deterministic ODE methods [16–19, 32]. We instead formulated a reaction model corresponding to the reactions illustrated in Fig. 1 in the form of rules specified in the BioNetGen language [38]. We used Hill functions to describe the rate laws governing protein synthesis, including PTEN, MDM2, CyclinD (CD), Myc, E2F and CyclinE (CE). Our choice was motivated by several studies [19, 39–41], which showed that transcription rates of these proteins are sigmoidal functions of transcription factor (TF) concentrations with positive cooperative Hill coefficients. We used mass action rules for other types of chemical reactions. Both ODEs and Gillespie's stochastic simulation algorithm (SSA) [42] are used to simulate the model with BioNetGen [38]. Stochastic simulation is important because when the number of molecules involved in the reactions is small, stochasticity and discretization effects become more prominent [43–45]. In the online Additional file 1, we list 23 ordinary differential equations which describe the deterministic HMGB1 model and all the input parameters. The BioNetGen code which implements SSA and ODE models is available at [46].

Since our understanding of many chemical reactions at the mechanistic level is not clear, a large number of parameters involved in these reactions are difficult to estimate based on existing data. We emphasize that in our HMGB1 model the values for some undetermined parameters were chosen in order to produce a qualitative agreement with previous experiments.

Model Checking

Model Checking [22, 47] is one of the leading techniques for the automated verification and analysis of hardware and software systems. Given a high-level behavior specification, a model checker verifies whether a system (or model) satisfies it. A specification might be satisfied by many different models. Thus, model checking is the process of determining whether or not a given system model satisfies (is a model of) a property describing the desired behavior of the system. Mathematically, system models take the form of state-transition diagrams, while some version of temporal logic [48] is used to describe the desired properties (specifications) of system executions. A typical property stated in temporal logic is G(grant_req→ F ack), meaning that it is always (G = globally) true that a grant request eventually (F = future) triggers an acknowledgment. One important aspect of Model Checking is that it can be performed algorithmically - user intervention is limited to providing a system model and a property to check.

The Probabilistic Model Checking problem (PMC) is to decide whether a stochastic model satisfies a temporal logic property with a probability greater than or equal to a certain threshold. To express temporal properties, we use a logic in which the temporal operators are equipped with bounds. For example, the property "CyclinD will always stay below 10 in the next fifty time units " is written as G⁵⁰(CyclinD < 10). We now ask whether our stochastic system M satisfies that formula with a probability greater than or equal to a fixed threshold (say 0.9), and we write M |= Pr_{≥ 0.9}[G⁵⁰(CyclinD < 10)]. In the next section, we formally define the temporal logic used in this work, Bounded Linear Temporal Logic [23].

Statistical Model Checking

We briefly explain Statistical Model Checking [50, 51], the technique we use for verifying BioNetGen models simulated by Gillespie's algorithm. Statistical Model Checking treats the Probabilistic Model Checking problem as a statistical inference problem, and solves it by randomized sampling of the traces (simulations) from the model. In particular, the PMC problem is naturally phrased as a hypothesis testing problem, i.e., deciding between two hypotheses - M ⊨ Pr_≥θ[ϕ] versus M ⊨ Pr _{< θ} [ϕ]. In other words, to determine whether a stochastic system M satisfies ϕ with a probability p ≥ θ, we test the hypothesis H₀ : p ≥ θ against H₁ : p < θ. Sampled traces are model checked individually to determine whether a given property ϕ holds, and the number of satisfying traces is used by a hypothesis testing procedure to decide between H₀ and H₁. Note that Statistical Model Checking cannot guarantee a correct answer to the PMC problem. However, the probability of giving a wrong answer can be made arbitrarily small.

We have introduced a Bayesian sequential hypothesis testing approach and applied it to the verification of rule-based models of signaling pathways and other stochastic systems [23, 49]. Sequential sampling means that the number of sampled traces is not fixed a priori, but is instead determined at "run-time ", depending on the evidence gathered by the samples seen so far. This often leads to a significantly smaller number of sampled traces.

Suppose that the stochastic system M satisfies the BLTL formula ϕ with some (unknown) probability p. The key idea behind statistical model checking [50] is that the behavior of M (with respect to property ϕ) can be modeled by a Bernoulli random variable with success parameter p. Such a random variable can be repeatedly evaluated via system simulation in the following way. Let σ be a trace of M, then the Bernoulli random variable X with (conditional) probability mass function:

$\begin{array}{cc}f(x|p)=p^x{(1-p)}^{1-x}& x\in \{0,1\}\end{array}$

(1)

denotes the outcome of σ ⊨ ϕ (i.e., model checking ϕ on σ). In other words, we have that:

$X=\{\begin{array}{ll}1\text{with probability}p\hfill & (\sigma |=\varphi ),\hfill \\ 0\text{with probability}1-p\hfill & (\sigma |=\neg \varphi ).\hfill \end{array}$

(2)

Therefore, by running a system simulation (i.e., a BioNetGen stochastic simulation) and by checking ϕ on the resulting trace we can obtain a sample from random variable X. When a sample of X evaluates to 1 we call it a success, otherwise, a failure.

Recall that in hypothesis testing we decide between a null hypothesis H₀ and an alternative hypothesis H₁:

$\begin{array}{cc}{H}_0:p\ge \theta & H_{\text{1}}:p<\theta .\end{array}$

(3)

The Bayesian approach assumes that p is given by a random variable whose distribution is called the prior distribution. The prior is usually based on our previous experiences and knowledge about the system.

Since p is a probability, we need prior distributions defined over [0,1]. In particular, Beta priors are mathematically convenient to use. They are defined by the following probability density:

$\begin{array}{cc}\forall u\in [0,1]& g(u,\alpha ,\beta )=\frac{1}{B(\alpha ,\beta )}{u}^{\alpha -1}{(1-u)}^{\beta -1}\end{array}$

(4)

where the Beta function B(α, β) is defined as:

$B(\alpha ,\beta )={\displaystyle {\int }_0^1{t}^{\alpha -1}}{(1-t)}^{\beta -1}dt.$

(5)

For later use, the Beta distribution function F_(α;β)(u) of parameters α, β is defined as for all u ∈ [0, 1] as:

$F_{(\alpha ,\beta )}(u)={\displaystyle {\int }_0^{u}g}(t,\alpha ,\beta )dt$

(6)

$=\frac{1}{B(\alpha ,\beta )}{\displaystyle {\int }_0^u{t}^{\alpha -1}}{(1-t)}^{\beta -1}dt.$

(7)

Let d = (x₁,..., x _n ) denote n samples of the Bernoulli random variable X defined by (2). Let H₀ and H₁ be the hypotheses in (3), and suppose that the prior probabilities P (H₀) and P (H₁) are strictly positive and satisfy P (H₀) + P (H₁) = 1. By Bayes's theorem, the posterior probabilities of H₀ and H₁, with respect to data d, are:

$\begin{array}{cc}P(H_i|d)=\frac{P(d|H_i)P(H_i)}{P(d)}& (i=0,1)\end{array}$

for every d with P (d) > 0. In our case, P (d) is always non-zero (there are no impossible finite sequences of data). The hypothesis test method is based on the Bayes Factor, that is, the likelihood ratio of the sampled data with respect to the two hypotheses. The Bayes Factor ℬ of sample d and hypotheses H₀ and H₁ is

$ℬ=\frac{P(d|H_0)}{P(d|H_1)}$

and by Bayes' theorem, we have that:

$ℬ=\frac{P(H_0|d)}{P(H_1|d)}·\frac{P(H_1)}{P(H_0)}.$

(8)

Therefore, B can be interpreted as a measure of evidence (given by the data d) in favor of H₀. Now, fix a threshold T > 1. The algorithm iteratively draws independent and identically distributed (iid) sample traces in the form of BioNetGen stochastic simulations, and checks whether they satisfy ϕ (Note that BioNetGen ensures by construction that each simulation, or trace, is actually iid.) After each trace, the algorithm computes the Bayes Factor B to check if it has obtained conclusive evidence. The algorithm accepts H₀ if B >T, and rejects H₀ (accepting H₁) if $ℬ<\frac{1}{T}$. Otherwise $\left(\frac{1}{T}\le ℬ\le T\right)$, it continues drawing iid samples. The statistical Model Checking algorithm is shown in Figure 2.

The following Proposition shows that, in our special case of Bernoulli samples, the computation of the Bayes Factor is straightforward.

Proposition 1. [49]The Bayes Factor of H₀ : p ≥ θ vs. H ₁ : p <θ with Bernoulli samples (x₁,..., x _n ) and Beta prior of parameters α, β is:

${ℬ}_n=\frac{P(H_1)}{P(H_0)}·\left(\frac{1}{F_{(x+\alpha ,n-x+\beta )}(\theta )}-1\right)$

where $x={\displaystyle {\sum }_{i=1}^n{x}_i}$ is the number of successes in(x₁,..., x _n ) and F_(s,t)(·) is the Beta distribution function of parameters s, t.

The Beta distribution function can be efficiently computed by standard software packages. Thus, no numerical integration is required for the evaluation of the Bayes Factor.

Finally, we must show that the error probability of our decision procedure, i.e., the probability that we reject (accept) the null hypothesis although it is true (false), can be bounded.

Theorem. [49]The error probability for the sequential Bayesian hypothesis testing algorithm is bounded above by $\frac{1}{T}$where T is the Bayes Factor threshold given as input.

Simulation results

The baseline stochastic simulations in Fig. 3A demonstrate that the expression levels of p53 and MDM2 _p oscillate even after 10 hours, when the cell enters the S phase (recall that cells usually remain in phase G1 for about 10 hours before moving to the S phase). However, oscillations are strongly damped in the ODE simulations (Fig. 3C) when the cell proceeds to the S phase, approximately after 10 hours. The stochastic simulation model is thus more consistent with the experimental results of Geva-Zatorsky et al. [52]. In that experiment the authors measured the dynamics of p53 and MDM2 _p in human breast cancer cells damaged by γ radiation. It was observed that the oscillations of p53 and MDM2 _p expression levels can last more than 72 hours after irradiation.

Fig. 3B and 3D show that the CyclinE protein, which regulates the G1-S phase transition in the cell cycle, reaches its maximum at about 10 hours, after which the cell proceeds with DNA replication (S phase). How does the expression level of HMGB1 and other proteins influence the cell's fate? We varied the levels of HMGB1 and AKT to determine how they affect cell behavior. A number of studies have found that HMGB1 is overexpressed in many cancers, and the overexpression of HMGB1 and its receptors can promote cancer cell proliferation and decrease apoptosis [8, 9]. In Fig. 4A-B, we increase the initial values of HMGB1 from 1 to 10⁶ and measure p53's maximum expression level in phase G1. We then measure the oncoproteins E2F and CyclinD/E's expression levels at 10 hours, which corresponds to the G1-S phase transition point. For the stochastic simulation, the experiment is repeated 10 times per value to compute the mean and standard errors. Fig. 4(A, D) demonstrates that the increase of HMGB1's initial value will lead to a decrease of p53's expression level, but when the number of HMGB1 molecules is over 10⁵, p53 will not continue to decrease. This is because HMGB1 can also activate and increase the expression level of its downstream protein E2F (Fig. 4(A, D)), whose overexpression will activate the transcription of the tumor suppressor protein ARF, which can inhibit MDM2's activity to stabilize p53's level. However, ARF is found to be mutated in up to 80% of pancreatic cancers [36, 53]. This means that ARF cannot inhibit the activity of the oncoprotein MDM2, thereby leading to lower levels of the tumor suppressor p53.

Fig. 4(B, E) shows that the cell cycle regulatory proteins CyclinD/E will increase with the elevated expression of HMGB1, a behavior which could be verified by future experiments. Fig. 4(A-B, D-E) explains the experimental discovery that the overexpression of HMGB1 decreases apoptosis and promotes DNA replication and proliferation in cancer cells.

The oncoprotein AKT is overexpressed in many cancers [54]. In Fig. 4(C, F), we first increase the number of unphosphorylated AKT molecules and fix the other proteins' concentration, then measure p53 and MDM2 _p 's expression levels at 10 hours in phase G1 after HMGB1 activates its receptor RAGE. Fig. 4(C, F) shows that with the increase of AKT's expression level, p53 is repressed due to the ubiquitination initiated by the overexpressed MDM2 _p , which is promoted by the activated and overexpressed AKT protein. The results in Fig. 4(C, F) suggest a way to inhibit tumor cell proliferation and induce tumor cell apoptosis through the inhibition of protein phosporylation events downstream from AKT kinases in the PI3K/AKT pathway, using an AKT kinase inhibitor (such as the drug GSK-690693 [55]).

K-RAS is a member of the RAS protein family. K-RAS mutation and ARF loss occur in more than 80% of pancreatic cancers [36, 53]. The P21 and FBXW7 proteins are also frequently mutated in many cancers [37]. ARF and P21 play an important role in the crosstalk between the p53 and Rb pathways. ARF is able to reroute cells with oncogenic damage to p53-dependent fates through binding to MDM2 and targeting its degradation. The p53-dependent tumor suppressor proteins P21 and FBXW7 can inhibit CyclinD/E's activity to prevent the proliferation of cancer cells.

Fig. 5 shows how mutations of ARF, P21 and FBXW7, and K-RAS influence tumor suppressor and cell cycle regulatory protein levels at 10 hours in the HMGB1 signaling pathway. We use the MDM2 degradation rate driven by ARF, d _ARF (${d^{\prime }}_7$ in the ODE model), to describe ARF mutations. Also, we use the Cyclin degradation rate driven by P21 (d_{P 21}for stochastic simulation, and ${b^{\prime }}_6$ for ODE simulation) to describe P21 and FBXW7 mutations. Large d _ARF and d_{P 21}values correspond to small mutations of ARF and P21 respectively, while small d _ARF and d_{P 21}values correspond to large ARF and P21 mutations in the cell.

Fig. 5(A, D) shows that wild-type ARF (large d _ARF ) can decrease the number of MDM2 _p molecules and increase p53's expression level to initiate apoptosis even if the cell proceeds to the S phase. Moreover, mutated ARF (smaller d _ARF ) can not stabilize p53 expression and prevent the proliferation of cancer cells if HMGB1 is overexpressed. This could explain the phenomenon that ARF loss exists in over 80% of pancreatic cancers [36]. Fig. 5(B, E) demonstrates that CyclinD/E proteins will increase if P21 is mutated (smaller d_{P 21}), thereby accelerating cell cycle progression.

K-RAS is mutated in most cancers, especially in pancreatic cancer [31]. The activation of RAS is initiated by HMGB1 and its receptors, and the wild-type RAS can be deactivated by some kinases. Studies have found that the mutated K-RAS can not be deactivated [56], even if HMGB1 is knocked out, so it will continuously activate the downstream signaling pathways which promote cell proliferation. Fig. 5(C, F) shows that with the increase of RAS deactivation rate d _RAS (b₁ in the ODE model), the synthesis of CyclinD/E will be inhibited, but a small deactivation rate of RAS will lead to overexpression of CyclinD/E. The results visualized in Fig. 5 suggest some ways to inhibit cancer cell proliferation through inhibition or deactivation of the signaling pathways involving RAS, Cyclin, and Cyclin-dependent kinases (CDK). Recently, CDK and RAS inhibitor drugs [57–59] have been developed to inhibit tumor growth.

Verification of the HMGB1 model

We use Statistical Model Checking (SMC) to verify some fundamental properties that our model should satisfy. We test whether the model satisfies a given BLTL property with probability p ≥ 0.9. We set the threshold T = 1000 for the verification, so the probability of a wrong answer is smaller than 10^-3.

that is, within 100 minutes p53's level will eventually be larger than 5.3 × 10⁴. SMC accepts this property as true, after sampling 38 traces (of which 37 satisfying traces).

with different RAS deactivation rates (d _RAS ). The results are presented in Table 4. Properties 4 and 5 show that the overexpression of HMGB1 and mutation of RAS (small d _RAS value) will accelerate the expression of cell regulatory protein CyclinD/E to promote cell proliferation. However, inhibition of HMGB1 and an increase of RAS deactivation rate will prevent tumor growth.

SMC accepted this property as true (22 satisfying traces).