Communications: Hamiltonian regulated cell signaling network

Abstract

Cell signaling is fundamental to cell survival and disease progression. Traditional approaches to study these networks have focused largely on probabilistic approaches, with a large number of ad hoc assumptions. In this paper, we develop a linear Hamiltonian model to study the integrin signaling network. The integrin signaling network is central to cell adhesion, migration, and differentiation, but has not been studied in the same detail as other cell cycle networks. In this study, the integrin signaling network with 16 nodes in thermal fluctuations is analyzed through ensemble averages on the linear Hamiltonian model. This new and analytically rigorous approach offers a quick method to find out the dominant nodes in the complex network, which operate in the thermal noise regime. The robust on∕off transitions due to the different initial inputs also reflect the inherent structure in the network, providing new insights into structure and function of the network.

INTRODUCTION

In recent years, a large number of signaling networks have been characterized through various tools rooted in statistical mechanics, graph theory, and probabilistic models.^{1, 2, 3, 4, 5, 6} Conventional approaches, however, often suffer due to lack of critical information such as reaction rates and chemical concentrations.^{7, 8} Another approach often utilized to study biological networks is based on probability distributions of likely states. A key limitation of this model is that it is often convergent to one unique final steady state irrespective of the initial conditions. As a result, cellular networks, such as integrin signaling that are sensitive to initial conditions cannot be accurately modeled with these probabilistic approaches.^{9, 10}

A key quantity of interest in biological networks is the robustness of these networks.^{10, 11, 12, 13} A robust network stabilizes the physical and chemical properties even when network information changes significantly, which gives the mechanism of fault tolerance in the signaling network.^{14, 15} Nevertheless, many signaling networks are required to be tunable in order to adapt in the different physiological environments. In order to address the limitations of probabilistic and chemical models and understand robustness of integrin signaling network, we provide an alternative approach that integrates the key aspects of the two models into a single framework of a Hamiltonian. This allows us to address key issues in signaling networks through analytic tools and adapt our approach to various forms of networks of varying sizes.

HAMILTONIAN FOR CELL SIGNALING NETWORK

The interactions within the network are of central importance in describing the dynamical behaviors of cell signaling through the corresponding Hamiltonian. In this paper, the key physical properties of the integrin signaling network are characterized by an interacting Hamiltonian. Here, we consider that each node i in the signaling network could only be on the state, either fully on (1) or off (0). The concentration c demonstrates the possibilities of each node being either on or off at the distinct biological period. The rules and constraints are inherent and consistent in the Hamiltonian framework:

• (i)Dependences of real Hamiltonian H: connectivity matrix W and concentrations c₀ and c_f of prohibitors∕catalysts at initial and final states yield H(W,c₀,c_f).
• (ii)Commutative rule: The flipping between initial and final states opposites the sign of Hamiltonian, H(W,c₀,c_f)=−H(W,c_f,c₀).
• (iii)Transitivity rule: Hamiltonian is a state function dependent on initial and final states only, i.e., H(W,c_i−1,c_i)+H(W,c_i,c_i+1)=H(W,c_i−1,c_i+1), where c_i=(1−λ_i)c₀+λ_ic_f is the intermediate state and λ_i is taken in the range of [0,1].

The Hamiltonian satisfying the rules is consistent with the free energy description, which is widely studied in the cell signaling network.^{10, 15} The simplest form of Hamiltonian can be constructed from the linear function of connectivity matrix W and concentration array c,

$$H\left(W,c_0,c_f\right)=\mu c_0^T\left(W-W^T\right)c_f,$$ (1)

where the initial concentration c₀ and final concentration c_f distribute over the entire network and the superscript T refers to transposition. Thus the linear component $\mu c_0^{T}W{c}_f$ weighs the interactions over the nodes except for the self-interaction, where μ provides the interaction strength. The weighted factors originate from the connectivity matrix W. This connectivity matrix defines “1” on (i,j) on off-diagonal elements when node i catalyzes the reaction in node j, “−1” when node i prohibits the reaction in node j, otherwise “0” when no direct interaction exists.⁹ However, the symmetry in matrix W does contribute nothing to the Hamiltonian because the same interactions in opposite directions cancel each other. We are therefore able to retain the antisymmetry part in the Hamiltonian formula by the rule (ii) above. The nonlinear effects such as self-interactions, non-neighbor interactions, and other higher order interactions in neighbor nodes, are assumed to be small perturbations to the linear model and are neglected in our current considerations. Typically, the chemical reaction speed in the signaling network is much faster than the macroscale time of the system. So the system approaches the equilibrium. The conditional probability predicts the output c_f based on the initial input pattern c₀,

$$P\left(c_f|c_0\right)={\frac{e}{Z}}^{-\beta H}=\frac{\text{exp}\left[-\beta \mu c_0^T\left(W-W^T\right)c_f\right]}{Z}.$$ (2)

Here, the partition function is Z=Σ_cfe^−βH. The β factor equals to 1∕k_BT for the temperature T and the Boltzmann constant k_B. Finally, we derive the probability distribution on the final state space written in terms of the average weighted on the initial states,

$$P\left(c_f\right)=\sum _{c_0}P\left(c_f|c_0\right)P\left(c_0\right)=\frac{e^{-\beta \Delta A}}{Z_0}=\frac{{⟨\text{exp}(-\beta \mu c_0^T\left(W-W^T\right)c_f\phantom{)}⟩}_0}{Z_0},$$ (3)

where Z₀ is the normalization constant and ΔA is the Helmholtz free energy dropping from the initial to the final state. The ensemble average⟨ ⟩ ₀ specifies that P(c_f) can be estimated by sampling only equilibrium configurations of the initial states. It also allows us to measure the initial probability on c₀ state space through experiments, and then update Eq. 3 iteratively.

FLUCTUATIONS OF INTERACTIONS

The simplest case considered in this paper is a system with no prior information about inputs. Therefore, the initial probability space is homogeneous, i.e., the equal distribution for each input pattern. We consider a network graph with N=16 nodes, representing a reduced integrin signaling network. Each node can have positive or negative influences on every other node. The network contains the nodes Abl, C3G, Cas, Crk, DOCK180, Graf, MEK, MLCK, MLCP, P190RhoGAP, PAK, RLC, ROCK, Rho, Src, and FAK, labeled in that order shown in Fig. 1. This network is based on previously published networks of integrin signaling.¹⁶

Figure 1.

The 16 nodes cell signaling network scheme. The relationship between each pair shows either the positive activating regulations (represented by arrow →) and the negative suppressing regulations (represented by a red cross ⊗).

Experimentally, the cell signaling network works in the thermal bath with the temperature. However, thermal fluctuations are considered as a universal phenomenon in the environment of all biological networks. Because the normal functioning cell signaling networks should be robust in their natural design in order to behave normally these thermal fluctuations. Specially, the complex functions and collective behaviors of the signaling network still make the properties of robustness unclear. In our study, the thermal fluctuations in the magnitude of k_BT are simulated through the random process in the interaction strength μ. The normal distribution with mean μ₀ and standard deviation σ_μ models the interaction strength μ including the thermal fluctuations.

The weighted average on the homogeneous initial states from Eq. 3 yields the on∕off probability of each node in the final states. The results are collected in Table 1, where the intermediate number between 0 and 1 yields the on∕off probability of each node at final. There are three types of nodes classified depending on the standard deviation σ_p of the on∕off probability in the Table 1: determined nodes (DN, 0≤σ_p≤0.15), semidetermined nodes (SDN, 0.15≤σ_p≤0.25), and random nodes (RN, σ_p≥0.25). It shows the normally distributed μ not only randomizes the on∕off node probability but also distributes the standard deviation σ_p on nodes. The classification unveils the fact that DNs are on∕off with small uncertainties, SDNs have high on∕off probabilities with large uncertainties and RNs are on∕off in a random fashion (Fig. 2). Within our expectation, the low thermal fluctuation (0.1%) is less likely to disturb the network system no matter what the node is. But the DNs still work well even when the physiological environment increases the thermal fluctuation up to 100%. In our results, the thermal noise can affect the certain nodes of signaling network and survive the others even in medium and high noise environment. Actually, it reflects the inherent characteristics of the complex network under the thermal fluctuation. The DNs should be in charge of the main functions of this integrin signaling network and the SDNs play important roles in the cycles of the biochemical process. Nevertheless, the RNs absorb the shocks from the thermal fluctuations and improve the total robustness of the system.

Table 1.

The 16 nodes network is strongly robust even in the very chaotic thermal fluctuations. When the thermal fluctuation vary from 5% to 100% comparing to the mean value of the interaction strength μ₀=10 (normalized in k_BT), the deviations of nodes only change in a narrow range. So the classification of nodes is unchanged.

σμ∕nodes	FAK	Src	P190RhoGAP	Rho	Rock	MLCP	RLC	MLCK	PAK	MEK	C3G	Cas	Graf	Crk	Abl	Dock180
0.01	0	0	0	0.06	0.05	0.09	0	0.04	0	0	0	0	0	0	0	0.05
0.5	0	0.06	0	0.12	0.21	0.43	0.31	0.41	0	0.22	0.06	0	0	0	0	0.2
1	0.11	0.11	0	0.17	0.21	0.38	0.33	0.38	0.01	0.23	0.11	0	0	0.12	0	0.24
5	0.11	0.15	0	0.18	0.24	0.45	0.44	0.43	0.15	0.24	0.26	0	0	0.11	0.11	0.3
10	0.12	0.12	0	0.11	0.21	0.4	0.39	0.41	0.15	0.23	0.15	0.1	0	0	0	0.25
Types	DN	DN	DN	SDN	SDN	RN	RN	RN	DN	SDN	SDN	DN	DN	DN	DN	SDN

Figure 2.

The heat map shows the inherent structures of the integrin signaling network with parameters μ₀=10 and σ_μ=1. The vertical column on the corresponding node illustrates the possibility distribution between on(1) and off(0) states. The DNs, SDNs, and RNs are distinct by their statistical characteristics.

It is worthy showing the new Hamiltonian description is consistent with the frame work of free energy. We considered the nodes that are on for the state based on the bit pattern of the state index, and summed the network influence for these N nodes, so the concentrations of the final state $c_f={\sum }_{i=0}^{2^N}{a}_i{P}_i$ expanded in the 2^N dimensional bitwise state space, where N is the number of nodes in the signal network. P_i is the N dimensional basis in the binary expansion of the corresponding state (each element in P_i takes either fully 1 or off 0 depending the corresponding nodes is on or off). The expanding coefficient a_i represents the possibilities of the binary state P_i. Then the free energy as a key function of state in the isothermal system is written in

$$\frac{A}{k_{B}T}=\sum _{i=1}^{2^N}-\text{ln}\left(a_i\right)+a_i\text{\hspace{0.17em}ln}\left(a_i\right),$$ (4)

where the summation is over the bitwise state space.

Figure 3 demonstrates free energy in the noisy thermal environment. The magnitude of free energy falls in the reasonable range of a previous study,¹⁰ which uses the free energy description instead of Hamiltonian. In the biological activity, ATPs always serves as the external energy source and transport the energy into the network system. Under typical cellular conditions, ΔA for the hydrolysis of ATP is approximately −57 kJ∕mol (−22.1 kT∕ATP). Therefore, more ATP molecules are required to maintain the normal functions of the cell signaling network system in the much noisy environment.

Figure 3.

The free energy curve is normalized by k_BT in the thermal fluctuation. The error bars shows the standard deviations due to the thermal noises.

SIGNIFICANT INPUTS

From a practical perspective, mechanics of cell signaling networks can provide clues to two fundamental questions, i.e., can the inputs control the final output results and which inputs significantly regulate the outputs? As shown in the previous section, the determined nodes are responsible for the main biochemical functions in integrin signaling network. Therefore, the proper choices of the initial on∕off states for determined nodes can significantly affect the network performance and output in the framework of Hamiltonian. But the previous models based on the free energy description fail to depend on the choices of the initial states. To illustrate this point, we explore the determined nodes FAK and Src in four different arrangements of initial inputs: (FAK:off, Src:off), (FAK:on, Src:off), (FAK:off, Src:on), and (FAK:on, Src:on). It is observed that the Cas node transits from off state to on state depending on the significant arrangement of input (FAK, Src) in the Fig. 4. Interestingly, the transition dynamics are quite stable in the thermal fluctuations. So the cell signaling network in vivo is not only robust in the equilibrium state but also in the transitions because the random nodes can absorb the external noises (see Table 1).

Figure 4.

The off∕on transition for node Cas due to the different input states on FAK and Src. Five sets of data illustrate the thermal fluctuation effects.

CONCLUSION

In this study, we have constructed the linear form of Hamiltonian based on the fundamental properties of interaction mechanics in the cell signaling network and have applied it to integrin signaling network. The statistical behavior of the linear Hamiltonian model is strongly nonlinear, and our model allows us to quantitatively describe the robustness of such a system in the presence or absence of thermal fluctuations. The real value of our model, however, lies in quickly identifying the dominant nodes that are in charge of the main functions in the biochemical reactions. The SDNs and RNs also play critical roles, such as intermediate products and noise absorbers. The on∕off transition on DNs represents the signaling network that can control by external inputs on specific nodes but suppress the unexpected changes due to the thermal noise. Comparing with the free energy models, the framework of Hamiltonian is successful to explain the effects on the choice of the initial states. Together, these results offer the possibility of improved understanding of the integrin signaling network and can serve to fundamentally understand numerous physiological processes regulated by similar networks.