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No authors have competing interests.

We show that logic computational circuits in gene regulatory networks arise from a fibration symmetry breaking in the network structure. From this idea we implement a constructive procedure that reveals a hierarchy of genetic circuits, ubiquitous across species, that are surprising analogues to the emblematic circuits of solid-state electronics: starting from the transistor and progressing to ring oscillators, current-mirror circuits to toggle switches and flip-flops. These canonical variants serve fundamental operations of synchronization and clocks (in their symmetric states) and memory storage (in their broken symmetry states). These conclusions introduce a theoretically principled strategy to search for computational building blocks in biological networks, and present a systematic route to design synthetic biological circuits.

We show that the core functional logic of genetic circuits arises from a fundamental symmetry breaking of the interactions of the biological network. The idea can be put into a hierarchy of symmetric genetic circuits that reveals their logical functions. We do so through a constructive procedure that naturally reveals a series of building blocks, widely present across species. This hierarchy maps to a progression of fundamental units of electronics, starting with the transistor, progressing to ring oscillators and current-mirror circuits and then to synchronized clocks, switches and finally to memory devices such as latches and flip-flops.

In all biological networks [

Functional building blocks should offer computational repertoires drawing parallels between biological networks and electronic circuits [

In a previous work [

Thus, our constructive theoretical framework identifies 1) building blocks of genetic networks in a unified way from fundamental symmetry and broken symmetry principles, which 2) assure that they perform core logic computations, 3) suggests a natural mapping onto the foundational circuits of solid state electronics [

We start our analysis by considering the dynamics of the most abundant network motif in transcriptional regulatory networks, the so-called feed-forward loop (FFL) introduced by Alon and coworkers [

A: FFL network representation. B: Numerical solution of FFL dynamics. The expression levels of genes Y and Z do not synchronize. The oscillation pattern presented is due to the square-wave behavior of gene X expression levels. We use _{x} = 0.12, _{y} = 0.7, _{x} = 0.5, _{y} = 0.1, _{0} = 0.7 and _{0} = 0.0. C: Input tree representation of FFL. The input trees of genes X, Y, and Z are not isomorphic, as a consequence, their expression levels do not synchronize. D: Base representation of FFL. The _{x} = 0.775, _{y} = 0.775, _{x} = 0.5, _{y} = 0.1, _{0} = 0.85 and _{0} = 0.0. Again, the oscillation is due to the wave-like pattern of X. G: As a result, genes Y and Z have isomorphic input trees. However, the input tree of the external regulator

We illustrate this result by presenting an analytical solution of the FFL [_{t} and _{t} of genes Y and Z, respectively, as a function of time _{t} is the expression level of gene X, _{x} and _{y} are the strength of the interaction representing the maximum expression rate of genes X and Y, respectively, and the thresholds _{x} and _{y} are the dissociation constant between the transcription factor and biding site. The expression level is measured in terms of abundance of gene product, e.g., mRNA concentration. The Heaviside step functions _{t} − _{x}) and _{t} − _{y}) represent the activator regulation from gene X and Y, respectively. They represent the Boolean logic approximation of Hill input functions in the limit of strong cooperativity [

In _{t → ∞}(_{t} − _{t}) ≠ 0. For example, _{0} > _{y} and _{x}. Specifically, we use the parameters: _{x} = 0.12, _{y} = 0.7, _{x} = 0.5 and _{y} = 0.1. For this combination, _{t} and _{t} do not synchronize since _{t} saturates at _{t} → _{x}/_{t} saturates at _{t} → _{x}_{y}/_{t} equal to a square wave and then monitor the expression levels of _{t} and _{t}. When _{x}, both _{t} and _{t} decay exponentially to zero. On the other hand, when _{x}, both variables evolve to saturate again at _{t} = _{x}/_{t} = _{x}_{y}/

In the FFL, gene Z receives input from X and Y, while gene Y, instead, only from X, and therefore the inputs are not symmetric (

For instance, by means of its autoregulation,

The FFF with activator regulations in

To understand the computation rationale of symmetric and frustrated circuits made of repressors, we map them to electronic analogues. We begin the analogy with the simplest circuit of a single gene with a feedback loop with repression (AR loop, _{t} is described by the discrete time model with Boolean interaction [_{y} is the maximum expression rate of gene Y, and _{y} is the dissociation constant.

A: A pnp transistor allows current flow if the voltage applied to its base is lower than the voltage at its emitter (_{B} < _{E}). Since it has a high (low) output for a lower (high) input, it is logically represented by a NOT gate. The yellow box shows the mapping between the pnp transistor and the biological repressor. B: A repressor regulation link plays the role of the pnp transistor since the rate of expression of a gene is repressed by gene Y if _{y} < _{t}. C: By connecting the base of the transistor to its collector, one forms a one stage ring oscillator. D: This connection is translated to the biological analogue as a repressor autoregulation at gene Y. In this way, the rate expression of gene Y is able to oscillate, depending on the adjustment of parameters _{y} and _{y} (see _{y}/_{y} > (_{x}/^{−1} (see _{x} = 0.454, _{y} = 0.454, _{x} = 0.5, and _{y} = 1.0.

The Heaviside step function _{y} − _{t}) reflects the repressor autoregulation in the Boolean logic approximation. We will show that this genetic repressor interaction, shown as the stub in

A transistor is typically made up of three semiconductors, a base sandwiched between an emitter and a collector (_{B} < _{E}) and thus the transistor acts as a switch and inverter. In the genetic circuit, the expression _{t} drives the rate of expression of gene Y, like the voltage drives current around an electric circuit. Simply comparing _{t} is an analogue for the base potential _{B} of a transistor, _{y} an analogue for _{E}, _{y} an analogue for the emitter current _{E}, _{t} an analogue for _{B}, and Δ_{t} an analogue for _{C}. Then, _{C} = _{E} − _{B} (

The repressor AR genetic circuit of

The UNSAT-FFF maps to the so-called Widlar current-mirror electronic circuit shown in _{t} = _{t} and oscillatory activity (

Next, we show analytically and numerically that the UNSAT-FFF circuit has an oscillatory solution plus synchronization of genes Y and Z.

The UNSAT-FFF circuit is obtained by the addition of a repressor AR loop to the FFL: in _{t} and _{t} are given by:
_{x} and _{y} are the strength of the interaction (maximum expression rate) of genes _{x} and _{y} are the dissociation constant, respectively. Similarly to the SAT-FFF case, synchronization between

We set λ = _{x}_{y}/_{y} _{t} = _{t}/_{y} and _{t} = _{t}/_{y} as
_{t} = _{x}, the solutions exponentially decay as _{t} = _{0}^{−t/τ} and _{t} = _{0}^{−t/τ}, where _{0} is the initial condition. For _{x},

We consider first the case where the initial condition is _{0} > 1. Thus, the solution of _{t} = _{0}^{−t/τ}, where ^{−1} = −log(1 − _{t} > 1, but ceases to be valid at a certain time _{t} < 1, which is given by _{0}⌉.

Next, we consider the case _{0} < 1. In this case the solution is given by _{t} = _{0}^{−t/τ} + λ(1 − ^{−t/τ}), which is always valid for λ < 1. Thus, when λ < 1 the system does not oscillate but it converges monotonically to a fixed point _{∞} = λ. However when λ > 1, this solution ceases to be valid at the time _{t} > 1. Therefore, the solution _{t} oscillates in time for λ > 1. For the case of _{0} > 1, the explicit solution is given by the general analytical expression which is plotted in _{0}) < 1 can be written in a similar way.

Thus, the main condition for oscillations in the circuits is λ > 1, and if λ < 1, there is no oscillatory behavior, and the solution _{t} converges monotonically to λ. Therefore, the oscillatory phase is separated from the non-oscillatory phase by the condition:

Thus, the repressor autoregulation at gene Y converts the circuit into a synchronized clock, a primary building block in any logic computational device.

Species | Database | Nodes | Edges | AR Fiber | FFF | Fibonacci Fiber | n = 2 Fiber | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

_{real} |
_{rand} ± |
Z-score | _{real} |
_{rand} ± |
Z-score | _{real} |
_{rand} ± |
Z-score | _{real} |
_{rand} ± |
Z-score | ||||

Arabidopsis Thaliana | ATRM | 790 | 1431 | 2 | 0.2 ± 0.5 | 4 | 2 | 0 ± 0 | Inf | 5 | 0.3 ± 0.6 | 8.1 | 0 | N/A | N/A |

Micobacterium tuberculosis | Research article | 1624 | 3212 | 11 | 0.7 ± 0.8 | 13.2 | 6 | 0.2 ± 0.4 | 14.6 | 4 | 1.7 ± 1.4 | 1.7 | 0 | N/A | N/A |

Bacillus subtilis | SubtiWiki | 1717 | 2609 | 35 | 0.3 ± 0.5 | 64.6 | 13 | 0.3 ± 0.5 | 23.4 | 1 | 1.3 ± 1.2 | -0.2 | 2 | 0 ± 0 | 63.2 |

Escherichia coli | RegulonDB | 879 | 1835 | 14 | 0.2 ± 0.5 | 29.1 | 12 | 0.1 ± 0.2 | 49.4 | 2 | 0.5 ± 0.8 | 1.9 | 1 | 0 ± 0 | > 3 |

Salmonella SL1344 | SalmoNet | 1622 | 2852 | 21 | 0.7 ± 0.8 | 25 | 14 | 0.2 ± 0.4 | 32 | 2 | 1.4 ± 1.3 | 0.5 | 3 | 0 ± 0 | > 3 |

Yeast | 10 | 5 | 3 | 0 | |||||||||||

YTRP_regulatory | 3192 | 10947 | 10 | 0.3 ± 0.6 | 17.3 | 4 | 0.2 ± 0.4 | 8.5 | 2 | 1.8 ± 1.3 | 0.2 | 0 | N/A | N/A | |

YTRP_binding | 5123 | 38085 | 2 | 0.1 ± 0.3 | 6.3 | 0 | N/A | N/A | 0 | N/A | N/A | 0 | N/A | N/A | |

Mouse | TRRUST | 2456 | 7057 | 1 | 0.1 ± 0.4 | 2.3 | 0 | N/A | N/A | 6 | 0.3 ± 0.6 | 9.3 | 0 | N/A | N/A |

Human | 1 | 1 | 100 | 1 | |||||||||||

TRRUST | 2718 | 8215 | 0 | N/A | N/A | 0 | N/A | N/A | 10 | 0.4 ± 0.6 | 16.3 | 0 | N/A | N/A | |

TRRUST_2 | 2862 | 9396 | 0 | N/A | N/A | 0 | N/A | N/A | 11 | 0.4 ± 0.7 | 16 | 0 | N/A | N/A | |

KEGG | 5164 | 59680 | 1 | 0.06 ± 0.25 | 3.76 | 1 | 0 ± 0 | > 3 | 79 | 0.6 ± 0.7 | 112 | 1 | 0 ± 0 | > 3 |

We report the Z-scores showing that all found fibers are statistically significant. We use a random null model with the same degree sequence (and sign of interaction) as the original network to calculate the random count _{rand} and compare with the real circuit count _{real} to get the Z-score.

The procedure to build more complex fibers can be systematically extended through an algebra of circuits that adds external regulators and loops to grow the

The addition of autoregulation loops and feedback loops results on a hierarchy of circuits of increasing complexity. For example, turning the A: repressor link into a B: repressor autoregulation results on an input tree that feeds its own expression levels with _{t} = 1 and branching ratio _{t} = 1, 2, 3, 5, 8, …. Here, gene X is not part of the _{t} = 2_{t−1} and branching ratio

From this starting point, one can grow the number of regulator genes, |1, _{t} representing the number of source genes with paths of length _{1} = 1), and _{t} represents the number of genes in the

A quantitative analysis of this measure yields exactly the golden ratio _{t} = _{t−1} + _{t−2} updating the current state two steps backwards, see [_{t−2} term in the Fibonacci sequence. This circuit has been synthetically implemented by Stricker

The complexity of the Fibonacci Fiber with feedback to the regulator is 1.6180…, which is lower than the number of loops in the circuit (two). The intuition of what this reveals is that, in this circuit the regulator X is still not part of the

All symmetric circuits shown in

We show next that static memory storage requires breaking the fibration symmetry of each circuit creating structures analogous to ‘flip-flops’ [

Next, we extend the constructive process described above to include symmetry breaking. We do so by mimicking an evolutionary process where circuits ‘grow’ by a ‘duplication’ event (analogous to gene duplication in evolution) that conserves the

The symmetry of the replica circuit is then explicitly broken by including different inputs, S and R, to regulate genes Y and Y’, respectively (

The symmetry is explicitly broken by applying set-reset inputs S ≠ R. Specifically, when S = 1 and R = 0, the circuit stores a bit of information. But here is the interesting fact: this state is kept in memory even when S = R = 1. That is, the circuit remains in the broken symmetry state even if the inputs are now equal and symmetric. In other words, the symmetry is now ‘spontaneously’ broken [

Thus, this genetic network is a toggle switch as studied in synthetic genetic circuits [

This spontaneous symmetry breaking [

Species | Database | Nodes | Edges | SR flip-flop | Clocked SR flip-flop | JK flip-flop | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

_{real} |
_{rand} ± |
Z-score | _{real} |
_{rand} ± |
Z-score | _{real} |
_{rand} ± |
Z-score | ||||

Arabidopsis Thaliana | ATRM | 790 | 1431 | 47 | 1.6 ± 1.2 | 36.40 | 3 | 0.2 ± 0.5 | 5.80 | 2 | 0 ± 0 | > 3 |

Micobacterium tuberculosis | Research article | 1624 | 3212 | 6 | 1.7 ± 1.4 | 3.20 | 0 | N/A | N/A | 0 | N/A | N/A |

Bacillus subtilis | SubtiWiki | 1717 | 2609 | 3 | 2.1 ± 1.4 | 0.6 | 0 | N/A | N/A | 0 | N/A | N/A |

Escherichia coli | RegulonDB | 879 | 1835 | 14 | 2.1 ± 1.4 | 8.40 | 3 | 0.3 ± 0.8 | 3.30 | 0 | N/A | N/A |

Salmonella SL1344 | SalmoNet | 1622 | 2852 | 6 | 1.4 ± 1.2 | 3.80 | 0 | N/A | N/A | 0 | N/A | N/A |

Yeast | 27 | 58 | 1 | |||||||||

YTRP_regulatory | 3192 | 10947 | 9 | 5 ± 2.5 | 1.60 | 3 | 3 ± 3.6 | 0 | 0 | N/A | N/A | |

YTRP_binding | 5123 | 38085 | 31 | 21.6 ± 5.8 | 1.60 | 192 | 103.3 ± 45.6 | 1.90 | 2 | 6.8 ± 6.1 | -0.8 | |

Mouse | TRRUST | 2456 | 7057 | 82 | 4 ± 2.1 | 37.70 | 216 | 1.9 ± 2.7 | 79.50 | 25 | 0.004 ± 0.06 | 417 |

Human | 192 | 566 | 90 | |||||||||

TRRUST | 2718 | 8215 | 89 | 4.3 ± 2.1 | 40.50 | 247 | 3.5 ± 4.8 | 50.60 | 45 | 0.02 ± 0.2 | 225 | |

TRRUST_2 | 2862 | 9396 | 103 | 5 ± 2.3 | 43 | 319 | 5.9 ± 7.2 | 43.20 | 45 | 0.02 ± 0.3 | 150 |

We report the corresponding Z-score statistics as computed in

Extending this duplication and symmetry breaking process to the FFF (

In summary, fibration symmetries and broken symmetries in gene regulatory networks reveal the functions of synchronization, clocks and memory through electronic analogues of transistors, ring oscillators, current-mirror circuits, and flip-flops. They result in a hierarchy of building blocks with progressively more complex dynamics obtained by iterating a procedure of replication and symmetry breaking. Beyond the circuits discussed here, the biological hierarchy can be extended to any number _{t} = _{t−1} + _{t−d}.

Gene regulatory structures are a mixture of combinational logic circuits, like FFF, and sequential logic circuits, like FF. They provide the network with a structure analogous to a programmable logic device or chip where the ‘register’ is a set of flip-flop circuits tied up together acting as the memory clock of the genetic network that feeds the combinatorial logic circuits made of simpler feed-forward circuit of low symmetry. Complex biological circuitry can then be seen as an emergent process guided by the laws of symmetry that determine biological functions analogous to electronic components. The discovery of these building blocks and building rules of logic computation will allow to: (1) systematically design synthetic genetic circuits following biological symmetry, and (2) systematically map the structure and function of all biological networks, from the symmetries of the connectome [

Detailed description of all analytical solutions mentioned in the main text, of the data acquisition and treatment, and detailed description of the proposed algorithm to find fibers.

(PDF)

In S2 File we present a list of circuits found in different species.

(PDF)

We are grateful to L. Parra and W. Liebermeister for discussions.