The Design Principles of Biochemical Timers: Circuits that Discriminate between Transient and Sustained Stimulation

Abstract

Many cellular responses for which timing is critical display temporal filtering-the ability to suppress response until stimulated for longer than a given minimal time. To identify biochemical circuits capable of kinetic filtering, we comprehensively searched the space of three-node enzymatic networks. We define a metric of "temporal ultrasensitivity," the steepness of activation as a function of stimulus duration. We identified five classes of core network motifs capable of temporal filtering, each with distinct functional properties such as rejecting high-frequency noise, committing to response (bistability), and distinguishing between long stimuli. Combinations of the two most robust motifs, double inhibition (DI) and positive feedback with AND logic (PF_AND), underlie several natural timer circuits involved in processes such as cell cycle transitions, T cell activation, and departure from the pluripotent state. The biochemical network motifs described in this study form a basis for understanding common ways cells make dynamic decisions.

Keywords: biochemical circuits; kinetic filtering; networks; signal transduction.

Figure 1.. Enumeration of 1-, 2-, and 3-node networks finds 25 minimal kinetic filtering circuits that distinguish between transient and sustained inputs.

(A) Kinetic filtering circuits respond to sustained but not transient stimuli, allowing cells to perform time-sensitive functions. (B) Temporal ultrasensitivity score quantifies kinetic filtering by measuring steepness of activation over stimulus time, defined by taking the ratio of input duration required for 10% activation to input duration required for 90% activation. Trigger time, the duration of input yielding 50% activation, measures the duration of stimulus necessary to trigger response. (C) To identify kinetic filtering architectures, temporal ultrasensitivity score and trigger time were measured over an enumerated space of 68,705 circuit topologies and 10,000 sampled parameter sets per topology testing with 36 specific input pulse durations between 0.25 and 50,000 seconds. Parameter sets were considered to show kinetic filtering if temporal ultrasensitivity score ≥ 0.5 and trigger time ≥ 1s. A topology’s robustness is defined as the fraction of its sampled parameter sets that show kinetic filtering. See Figure S1 for details on simulating enzymatic circuits with OR and AND nodes. (D) Number of topologies, kinetic filters with robustness ≥ 0.001, and minimal kinetic filters in 1-, 2-, and 3-node networks. Minimal kinetic filtering topologies are topologies with robustness ≥ 0.001 where removal of any link decreases robustness below 0.001 (Figure S2D). Two 2-node minimal kinetic filters are topologically identical to 1-node minimal kinetic filters with regulatory node B taking the place of the basal regulator. Circuits with robustness ≥ 0.01 are indicated in bold.

Figure 2.. Minimal kinetic filters of 1, 2, and 3 nodes phenotypically cluster into five groups.

(A) Metrics used to cluster minimal kinetic filters by phenotypic features. For metrics 2-5, a single input pulse of duration 50,000s was applied. Measurements of ON dynamics are relative to input ON time, and measurements of OFF dynamics are relative to input OFF time. OFF dynamics were not measured for circuits where max output = final output. Phenotypic metrics were measured for all parameter sets of minimal kinetic filters with temporal ultrasensitivity ≥ 0.5 and trigger time ≥ 1s (total of 2,896 parameter sets distributed across 25 topologies in Figure 1D). (B) Location of minimal kinetic filters in 3D space of first 3 principal components of 6 phenotypic metrics. See Figure S3 for singular values and composition of each principal component. Each sphere is centered at the mean principal component value observed over all kinetic filtering parameter sets of each minimal kinetic filtering architecture. Sphere size is proportional to radius capturing 60% of observed phenotypes. (C) Minimal kinetic filters cluster into 5 phenotypic groups that each share structural features. Archetypal topologies (right column) are the simplest topology in each phenotypic group. Circuit links where kinetic filtering requires Michaelis-Menten kinetics to be in the linear or saturated regime are indicated in blue and orange respectively.

Figure 3.. Representative timecourses, preferred parameter regimes, and robustness of five classes of kinetic filters.

Each circuit is shown responding to input shorter than trigger time, just longer than trigger time, and far longer than trigger time. Distributions of parameter values used to identify preferred regimes are shown in Figure S4. The dashed line in row B indicates the output concentration separating the attractor OFF and ON states. See Table S1 for parameter values used in example timecourses. Robustness score distributions across all topologies for each minimal kinetic filtering motif are shown. A topology was considered to contain a double inhibition motif if it contained at least one double inhibition minimal kinetic filter, and analogously for each of the other kinetic filtering classes. All topologies that do not contain motifs in any of the 5 classes have robustness < 0.001.

Figure 4.. Turning off a deactivator more effectively buffers against partial activation by subthreshold length inputs.

(A) Histogram of observed trigger times minimal kinetic filtering circuits. Parameter sets of minimal double inhibition topologies (#4, #21-24 in Figure 2C, total 798 parameter sets) and the coherent feed forward loop topology (#25 in Figure 2C, total 88 parameter sets) satisfying temporal ultrasensitivity score ≥ 0.5 and trigger time ≥ 1s were measured for trigger time. (B) An archetypal architecture for each kinetic filtering motif was sampled for 50,000 parameter sets over the same range as the sampling used in the enumerative search (k_cat 0.1 to 10, K_m 0.001 to 100, evenly in log space by Latin hypercube). Shown in each plot are the temporal ultrasensitivity score and trigger time for each parameter set of the archetypal topology that resulted in temporal ultrasensitivity score ≥ 0.3 and trigger time ≥ 10s (DI: 1328 parameter sets; CFFL: 1078 parameter sets; PFB_AND: 2135 parameter sets; PFB_OR: 115 parameter sets). (C) Steady state output changes at a more gradual pace with changing regulator concentration in double inhibition circuits compared to double activation circuits. In both circuits, we solved for steady state output node concentration as a function of [B] with K_mBC = 0.5, k_catBC = 1, K_m basal activator/deactivator = 0.5, k_cat basal activator/deactivator = 1, concentration of basal act./deact. = 0.1. (D) Positive feedback AND circuits are better kinetic filters than positive feedback OR circuits because they require output to remain low until feedback activation rises. Activation rate consists of activation due to input and activation due to feedback, which are multiplied in AND circuits and summed in OR circuits. Shaded region delineates the zone between 5% and 95% output activation.

Figure 5.. Natural examples of kinetic filters feature both core and combinatorial kinetic filtering motifs.

(A) Double inhibition, positive feedback AND and coherent feed forward loop AND form the core set of kinetic filtering motifs. (B) T cell activation is regulated by enzymatic networks that contain combined double inhibition and positive feedback AND motifs. (C) Pluripotent state exit in embryonic stem cells and cell cycle transitions in both mammalian and yeast cells are controlled by double inhibition and positive feedback AND architectures with mixed enzymatic and transcriptional regulation. (D) Erk activation of cFos is mediated by networks containing a coherent feedforward loop AND motif (E) Core kinetic filtering motifs can be combined to yield hybrid phenotypes.