Precritical State Transition Dynamics in the Attractor Landscape of a Molecular Interaction Network Underlying Colorectal Tumorigenesis

Abstract

From the perspective of systems science, tumorigenesis can be hypothesized as a critical transition (an abrupt shift from one state to another) between proliferative and apoptotic attractors on the state space of a molecular interaction network, for which an attractor is defined as a stable state to which all initial states ultimately converge, and the region of convergence is called the basin of attraction. Before the critical transition, a cellular state might transit between the basin of attraction for an apoptotic attractor and that for a proliferative attractor due to the noise induced by the inherent stochasticity in molecular interactions. Such a flickering state transition (state transition between the basins of attraction for alternative attractors from the impact of noise) would become more frequent as the cellular state approaches near the boundary of the basin of attraction, which can increase the variation in the estimate of the respective basin size. To investigate this for colorectal tumorigenesis, we have constructed a stochastic Boolean network model of the molecular interaction network that contains an important set of proteins known to be involved in cancer. In particular, we considered 100 representative sequences of 20 gene mutations that drive colorectal tumorigenesis. We investigated the appearance of cancerous cells by examining the basin size of apoptotic, quiescent, and proliferative attractors along with the sequential accumulation of gene mutations during colorectal tumorigenesis. We introduced a measure to detect the flickering state transition as the variation in the estimate of the basin sizes for three-phenotype attractors from the impact of noise. Interestingly, we found that this measure abruptly increases before a cell becomes cancerous during colorectal tumorigenesis in most of the gene mutation sequences under a certain level of stochastic noise. This suggests that a frequent flickering state transition can be a precritical phenomenon of colorectal tumorigenesis.

Fig 1. Flickering state transition before a critical transition in attractor dynamics.

(a) and (b) represent critical transitions without and with noise in the attractor dynamics, respectively. The x-axis represents the effector sequence, and the y-axis denotes the state of the system. A solid line indicates an attractor, and the dotted line between the two solid lines represents an unstable state. A critical transition to an alternative attractor state (A) occurs at a bifurcation point (F1 or F2). Effectors in (a) and (b) are factors changing the attractor landscape. (c), (d), and (e) indicate the attractor landscapes reflecting the stability properties of the system in the region of (X), (Y), and (Z), respectively. Because a potential on the y-axis is inversely related to the steady state probability of its state, the dynamics tends to converge to a state with lower potential. A ball (grey circle) represents the current state and its potential. (d) In the region of (Y), the ball jumps back and forth between alternative basins of attraction from the impact of noise, namely the flickering state transition ((B) in Fig 1(b)). Such a flickering state transition increases the variation in the estimate of the basin sizes (the sizes of the basin of attraction) for the attractors.

Fig 2. Flickering state transition during colorectal tumorigenesis in the conditions of N_I = 0.03.

Colorectal tumorigenesis is driven by the sequential accumulation of 20 gene mutations (the sequence of No. 73 in the S1 Table). Zero on the x-axis means no mutation. (a) and (b) the fraction of the initial states converging into the apoptotic, proliferative or quiescent attractors for 320,000 initial states at every gene mutation with N_I = 0 and 0.03, respectively. (c) A graphical representation of M_F (Eq (1)) to check the flickering state transition. B_A, B_P, and B_Q represent the fractions of the basin sizes for the apoptotic, proliferative, and quiescent attractors, respectively. B_A0, B_P0, and B_Q0 express the fractions of the basin sizes for the apoptotic, proliferative, and quiescent attractors in the absence of noise, respectively, and B_AN, B_PN, and B_QN are the fraction of the basin sizes for the apoptotic, proliferative, and quiescent attractors in the presence of noise, respectively. (d) M_F at every mutation occurrence for N_I = 0.03, as a result of Fig 2(a) and (b).

Fig 3. Generality of the more frequent flickering state transition before developing into colorectal cancer.

Colorectal tumorigenesis is driven by the sequential accumulation of 20 gene mutations for 100 representative sequences for 20 gene mutations that drive colorectal tumorigenesis (S1 Table). Zero on the x-axis means no mutation. (a) The frequency distribution of the occurrence point of the cancerous state along with the sequences that drove the cancerous state for the various levels of noise intensity. For the noise intensities of 0, 0.01, 0.02, and 0.03, the cancerous state occurs in 97, 99, 99, and 97 sequences of the 100 sequences, respectively (Table 1). (b) The frequency distribution of the M_F greater than the M_FTH at every mutation occurrence along with the sequences that drove the cancerous state for the various levels of noise intensity. We defined the upper threshold of M_F (M_FTH) to investigate whether the flickering state transition becomes more frequent in the attractor landscape. For the noise intensities of 0.01, 0.02, and 0.03, an M_F greater than the M_FTH appears in 46, 80, and 81 sequences of 99, 99, and 97 sequences that drove the cancerous state, respectively (Table 1). The y-axis indicates how many sequences among the sequences that drove the cancerous state have an M_F greater than the M_FTH at a particular mutation occurrence.