Fig 1.
Combinatorial logic bottlenecks information flow in networks.
(A) The number of ways that three TFs (K1, K2, K3) can be ON or OFF (tabulated at right) is the same as the number of ways they can bind at promoters (left). An equal number of gene expression states are observed whether the TFs use AND logic (requiring all factors be present) or OR logic (requiring at least one of the factors). (B) Signal-to-target information flow is bottlenecked by regulators if (i) the regulators respond to multiple targets, or (ii) the signals activate multiple regulators. The allowed target states are tabulated for signals using AND logic and regulators using AND/OR logic. (C) A feedback loop causes constitutive activation of a regulator (K1) and leads to fewer accessible configurations (tabulated at right).
Fig 2.
The ratchet model attains configurations not reachable by combinatorial logic.
(A) The ratchet model for n = m = 2 and ln = lm = 1. Activators (K’s) and deactivators (P’s) turn targets ON and OFF, respectively. (B) An example temporal sequence for the network with a threshold equal to 1. Black targets are in the 1 state. (C) An example sequence for the same network with threshold equal to 2. Gray targets are in the 1 state, and black targets are in the 2 state. (D) Scaling laws for the threshold T = 1 (red) and T = 2 (yellow) are shown for symmetric networks (n = m). A comparison to combinatorial logic with an equivalent number of regulators (n + m) is shown in blue.
Fig 3.
Multiple connections in the ratchet network decreases the number of configurations.
(A) An example network where each target has ln = 2 connections to the K’s (red) and lm = 2 connections to the P’s (blue). (B) A list of the minimal length sequences generating unique configurations in the network in when ln = lm = 1. Red bars are K actions and blue bars are P actions. (C) The list of minimal length sequences when ln = lm = 2. Some sequences now map to the same configuration. (D) Analytical solution for the number of sequences as a function of n = m for different ln = lm families.
Fig 4.
The ratchet network is robust to loss of a regulator.
(A) A schematic illustration of the experiment. The regulator K1 was deleted from networks with m = 2 P’s and variable n for different values of the connectivity ln. The resulting number of configurations was computed by simulation. (B) Correlation coefficient between configurations in the full network (all K’s; rows) and the impaired network (without K1; columns). All rows with exact matches were deleted. (C) Cumulative distribution F(x) of the maximum correlation coefficient x for each row in C for different values of ln. The dashed line is the similarity cutoff 0.8. (D) Tradeoff between reachability and robustness. The number of reachable configurations as a function of (n, ln) is plotted vs. the fraction of states above the similarity cutoff 0.8 (i.e. 1 − F(0.8)) for different values of n.
Fig 5.
The sequestration network is a noncommutative model of gene regulation by chromosome folding.
(A) A sequence of moves K3 K4 K1 P3 P4 on a hypothetical chromosome with K and P actions represented as DNA-binding factors and K1 playing the role of RNAP. Red circles correspond to genes and numbers correspond to allowed binding partners. (B) The same sequence in A represented as a collection of targets with up to n = 4 arms. For example, the target {0, 1, 2} corresponds to the gene locus with states 1 and 2 in A. The filled circle represents the current state.
Fig 6.
Scaling in the sequestration model is super-exponential.
(A) A plot of all the allowed configurations of a set of targets of n = 4 regulator pairs in the sequestration model. Yellow represents targets that are ON, and blue those that are OFF. (B) A list of the sequences generating the corresponding states in A. K actions are shown in the red spectrum, and P in the blue. (C) A logarithmic plot of the scaling in the sequestration model. The total space is the 22n − 1 − 1, the reachable space is calculated from Eq (7), and the combinatorial model is 22n.
Fig 7.
Noncommutative models induce orbits in the configuration space.
Graphical representation of the orbits in (A) the n = 4, m = 2 ratchet network and (B) the full n = 3 sequestration network. Configurations are indicated by red circles, and those accessible to each other are connected with blue lines. Arrows in (A) indicate whether a path is irreversible.
Fig 8.
Sequential logic on regulatory landscapes.
(A) The regulatory landscape for the 2-mRNA system X1, X2 for two hypothetical paths with configurations represented by balls. It is difficult to directly increase X2 because of a potential barrier (top). In the roundabout path (bottom), visiting two intermediate configurations via K1 K2 P1 results in an altered regulatory landscape. (B) The initial and final configurations in (A) projected onto (X1, X2) space (left) and (X1, X2, V1) space (right). The regulators affect not only X1 and X2, but also an additional variable, denoted V1, that alters the landscape of X1 and X2. The arrows indicate the instantaneous direction of the trajectory.