1.1. The threshold prediction problem

Classical approaches fail at different levels. Flux Balance Analysis (FBA) optimizes metabolic flux distributions but lacks regulatory control semantics [1]. Boolean network models capture genetic logic but treat metabolites as external parameters rather than dynamic variables [2]. Classical hybrid Petri nets separate metabolism (continuous dynamics) and regulation (discrete logic) into distinct sub-models without unified semantics for signal designation [3,4]. Biological Petri nets [5–7] with test arcs model catalytic regulation but cannot express signal consumption—the irreversible depletion creating commitment finality [8,9].

The expressiveness gap is fundamental: no existing Petri net formalism provides semantics for molecules that simultaneously transfer mass (metabolism) and broadcast information (regulation) through modeler designation, making threshold computation a structural rather than parametric property. Metabolites designated as regulatory signals must participate in biochemical transformations (normal arcs, mass transfer) and control pathway activation (regulatory arcs, information transfer) within unified model semantics. Classical approaches force separation: FBA ignores regulation, Boolean networks parameterize metabolites, hybrid Petri nets partition concerns into distinct sub-models. Without unified semantics enabling dual participation through signal type assignment, threshold values emerge only through experimental measurement or parameter fitting, never through topology-driven threshold computation.

1.2. Contributions

This work presents signal hierarchy theory and its computational realization as signal hierarchical Petri nets, enabling topology-driven threshold computation through four original contributions:

1. Signal Hierarchy Theory: We establish formal foundations for hierarchical biological control proving two structural theorems. The acyclicity theorem demonstrates that connected signal flow graphs (topology-explicit arcs defining commitment structure) must be directed acyclic graphs—formalizing commitment finality: once a decision is made, the cell must follow that biochemical path irreversibly without alternative routes back. The preemption theorem proves that lower-layer signal depletion structurally disables higher-layer transitions regardless of higher-layer token abundance—capturing the principle that resource exhaustion overrides regulatory programs. Critically, acyclicity ensures one-way commitment trajectories through connected channels, while feedback regulation occurs abundantly through remote sensing (rate functions ; formally defined in Section 3.2) without creating reversible commitment paths.
2. Signal Hierarchical Control: We introduce signal control theory establishing a graph of information layered over the graph of execution (Fig 1). By assigning signal type to selected places and arcs, we create vertical information channels broadcasting mass availability from Layer 0 (executive metabolic layer) to higher regulatory layers, while horizontal execution proceeds through stoichiometric transformations. Signal flow arcs implement consumptive commitment semantics distinct from test arcs (non-consuming catalysis): token consumption creates irreversible decisions partitioning state space through basin boundaries. Layer functions assign hierarchical organization, and preemption predicates formalize vertical dependencies where lower-layer depletion structurally disables higher-layer transitions.
3. Signal Hierarchical Petri Nets: We extend classical Bio-PN [10,11] from 5-tuple to 13-tuple hybrid formalism incorporating signal places (), signal flow arcs (), signal arc weights (), layer functions (), and enablement thresholds (). SHPN is a hybrid Petri net in the sense of supporting coexistent continuous and stochastic dynamics [4,12], but extends classical hybrid PN by adding unified signal-type semantics that resolve the sub-model separation limitation of prior approaches [3]: rather than partitioning metabolism and regulation into distinct sub-models, SHPN allows modeler-designated places to participate in both simultaneously. The formalism therefore supports both deterministic and stochastic simulation engines [13] while providing unified semantics for metabolic-regulatory integration. The extension preserves Bio-PN semantics while adding hierarchical capabilities: normal arcs for horizontal mass transfer, signal flow arcs for vertical information propagation. Modelers designate any metabolite (ATP, GTP, NADH, cAMP, Ca²⁺) as signal place when biological evidence supports regulatory gating role. Designated signals participate through both arc types simultaneously, unifying metabolic and regulatory roles within a single hybrid model.
4. Metabolic-Regulatory Unified Modeling: By unifying metabolic and regulatory networks through signal type assignment, commitment thresholds emerge from topology as structural properties computable via . Applied to B. subtilis sporulation, the formalism provides a mechanistic account of the Fujita and Losick [14] timing-dependent outcome asymmetry (abrupt , gradual ): the effective commitment threshold at M is identified empirically via simulation (Proposition 5), and only the gradual phosphorelay accumulation allows to ramp across it. This mechanistic prediction—inexpressible in classical Bio-PN lacking -threshold semantics—is quantified in Section 6.

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Fig 1. Dual-graph architecture of SHPN.

The execution graph (left) carries stoichiometric mass transfer through normal arcs (solid black) and may contain cycles. Signal places (blue hexagons, ) participate in both graphs simultaneously. The signal control graph (right) extracts those places into a directed acyclic graph (POSet); dashed arcs carry signal weight and basin-floor threshold , giving the commitment criterion . Layer depth encodes the hierarchy of commitment decisions. Acyclicity of formalizes biological irreversibility: a committed cell cannot uncommit through the same hierarchical channel. Cycles in remain fully unrestricted.

https://doi.org/10.1371/journal.pone.0355540.g001

Validation: Mathematical validation through proven theorems establishes correctness. Biological validation through B. subtilis case study demonstrates practical applicability, following established methods for Petri net pathway validation [15–17]. Together, these demonstrate that SHPN provides a structural account of biological commitment that is testable against independent experimental data and does not require global parameter fitting to reproduce experimental outcome ratios.

2.1. Fundamental concepts

Definition 1 (Signal Flow Graph). Let T be the set of transitions of the underlying Petri net. A signal flow graph consists of signal places (information channels) and signal flow arcs representing consumptive information propagation. Unlike metabolic arcs transferring mass, signal flow arcs transfer regulatory information consumed during transition firing.

Definition 2 (Layer Function). A layer function assigns hierarchical depth to signal places via topological sort of . For all signal flow edges, we enforce:

establishing directed information flow from lower (metabolic) to higher (regulatory) layers.

Definition 3 (Basin Boundary). For transition t with signal input arc , the commitment marking

defines basin boundary partitioning reachable state space:

• Below boundary: (transition t disabled)
• At boundary: (firing consumes , leaves residual )
• Above boundary: (firing optional)

Firing from produces when , creating one-way barrier: no firing sequence from B_below reaches B_above after signal consumption.

2.2. Structural theorems

Theorem 1 (Acyclicity Requirement). Signal flow graphs must be directed acyclic graphs (DAGs). Formally: admits valid layer function satisfying for all paths if and only if is acyclic.

Proof. () Assume valid layer function exists satisfying for all edges where t connects to via signal flow arcs. Suppose for contradiction that contains cycle

By layer function constraint applied to each edge:

Combining inequalities yields , contradicting irreflexivity of < on . Therefore must be acyclic.

() Assume is acyclic. We construct valid via topological sort algorithm:

1. Initialize for all
2. While exist unprocessed places: select p with indegree 0 in remaining graph
3. For each outgoing edge : set
4. Remove p from graph, update indegrees

Algorithm terminates because is finite and acyclic (no place depends on itself). For any edge , topological ordering ensures processed before , thus by construction. Hence valid layer function exists. □

Biological significance: Acyclicity ensures commitment finality—once a decision is made through connected hierarchical channels, there is no alternative path back. When a cell commits to sporulation, rises above its effective kinetic threshold M (via the rate function ), enabling T_septation to fire and consume GTP tokens irreversibly via the GTP_pool signal-flow arc—it must then follow that biochemical program irreversibly. The cell cannot “uncommit” and return to vegetative growth through the same signal flow path that triggered commitment. However, feedback regulation occurs abundantly through remote sensing (see Section 3.2): transcription factors can sense metabolite concentrations via rate functions without creating reversible commitment paths. The acyclicity constraint applies to topology-explicit connected paths (signal flow arcs defining irreversible commitment structure), not to equation-implicit remote observations (rate functions enabling feedback without reversibility). Signal consumption creates one-way state transitions: firing from basin boundary depletes signals, making reverse traversal impossible. This formalizes biological commitment: development proceeds forward through irreversible decisions, while metabolic feedback maintains homeostasis through remote sensing.

Theorem 2 (Hierarchical Preemption). Lower-layer signal depletion structurally disables higher-layer transitions, assuming the PreemptionCheck predicate is enforced: transition may fire only if every transition that produces its signal inputs via paths satisfies its local enablement conditions (NormalEnabled, TestEnabled, and SignalEnabled). Formally: if and there exists connected path through signal flow arcs, then

regardless of .

Proof. We prove by induction on layer depth .

Base case (k = 1): Places in adjacent layers with and . By layer function definition, there exists and . The SignalEnabled predicate for requires . If , then since , , hence . By definition, requires for every where . Since , we have for all . Hence .

Inductive step: Assume true for all layer gaps . Consider with . There exists intermediate place with on path . By inductive hypothesis applied to with gap : if , then for all producing via . Therefore for all (direct consequence of of ’s producers). By inductive hypothesis applied to : this propagates to all -producers of , hence for all . Transitive closure over the -acyclic path completes the proof. □

Biological significance: depletion below the commitment separatrix (M) structurally disables septation and all downstream execution layers, regardless of Spo0AP abundance. A cell with saturating phosphorelay output but below threshold cannot commit—the commitment gate overrides integration-layer readiness. This captures hierarchical override: Layer 2 preempts Layer 3 through threshold enforcement, preventing commitment without the required signal.

2.3. Basin boundary computability

Theorem 3 (Irreversible Basin Partition). For signal place with signal flow arc , assuming no other transition regenerates via (i.e., there is no with ), firing transition t from commitment marking creates a one-way barrier in the reachability graph. Formally, let:

where denotes reachable markings. Then firing t from any produces , and no firing sequence from B_below produces a marking in B_above.

Proof. Consider marking with . By signal enablement, holds with equality. Firing t consumes tokens from (by firing rule for signal places), producing:

If (no backward signal flow), then , placing .

For irreversibility, consider any with . To reach B_above, must increase by producing tokens through some transition with . By hypothesis, no such exists. Thus B_below is an absorbing set: once below the commitment threshold, no reachable firing sequence returns to B_above. □

Corollary 4 (Threshold Predictability). Commitment thresholds are computable from network topology without simulation:

where and derive from biological rate constants and stoichiometry.

Proof. Direct consequence of Basin Boundary Definition. The commitment marking is defined as the minimal marking enabling transition t such that firing consumes exactly the decision quota, leaving residual (the enablement threshold). By signal enablement condition SignalEnabled(t,M):

The minimal such marking is obtained by taking equality. Both parameters are formalism specifications: is assigned via threshold function , and via signal weight function . No simulation or state space exploration required—values obtained directly from model specification via arithmetic. □

This establishes threshold prediction as structural property of signal hierarchical networks. Unlike parameter fitting requiring experimental data, topology-based computation proceeds from formalism alone: identify signal input arc , extract threshold and weight , compute commitment marking. Below M_commit, transitions remain disabled; above, firing becomes possible but non-mandatory; at boundary, firing creates irreversibility.

3.1. Signal consumption semantics

The information graph operates through consumptive signal semantics distinct from classical arc types. Signal flow arcs implement information propagation where token consumption at Layer k triggers enablement re-evaluation at layers . This differs from:

Normal arcs (F): Model horizontal mass transfer with stoichiometric conservation. Firing consumes W((p,t)) inputs, produces W((t,p)) outputs, preserving molecular species through Layer 0 transformations.

Test arcs (): Model catalysis through non-consuming local observation. Require without token transfer, enabling rate acceleration via arc-connected catalysts. Test arcs are the appropriate model for catalysis on two independent grounds. First, they are the established formalism for read-only access in Petri nets (contextual/read arcs, [18]): they are semantically distinct from loops (W(p,t)=W(t,p)), which consume and regenerate the same quantity of tokens, because loops contribute to the stoichiometric firing rule while test arcs leave the marking unchanged — a semantic difference independent of concurrency. Second, test arcs correctly model biological catalysts: enzymes that enable reactions without being consumed must not contribute to mass balance, which loops would violate by introducing a spurious consume-then-regenerate cycle in the stoichiometric record.

Signal flow arcs (): Model vertical information broadcast with consumptive commitment. These arcs require marking and consume decision quota upon firing. Unlike horizontal mass conservation (normal arcs) or local catalysis (test arcs), signal consumption creates irreversibility—tokens represent commitment resources depleted upon firing, establishing one-way state transitions partitioning reachability space. The arc semantics couple token transfer with vertical broadcast to all layers .

The information graph thus layers hierarchical control over biochemical execution: signal places serve as interfaces where Layer 0 mass dynamics broadcast regulatory information vertically through consumptive arcs, while horizontal transformations proceed through conserved mass transfer.

3.2. Connected vs. Remote information access

The formalism distinguishes between topology-explicit (connected) and equation-implicit (remote) concentration information transfer:

Connected channels (Signal flow arcs ): Created by assigning type “signal” to normal arcs, coupling horizontal mass transfer with vertical information broadcast. Signal flow arcs are ordinary arcs to which we assign signal semantics: token transfer at Layer k (horizontal mass flow) triggers enablement re-evaluation at layers (vertical information broadcast). Signal places function as information hubs through this type assignment: the same places receive mass via metabolic reactions (normal arcs F) and broadcast availability via signal flow arcs . This creates an interface from Layer 0 (executive metabolic layer) to higher regulatory layers—mass flows horizontally through biochemical transformations, information about mass availability broadcasts vertically to the cellular regulatory environment. Arc topology makes regulatory dependencies explicit—connected paths define hierarchical control flow.

Test arcs implement non-consumptive observation of arc-connected places (local catalysis), distinct from remote sensing which operates through rate function global references without physical connection. Both connected channels and remote sensing provide concentration information: connected channels couple consumption with hierarchical broadcast through explicit topology for irreversible commitment, while remote sensing enables distance-independent regulation preserving local topology without preemption influence.

Remote sensing (Rate functions ): Implement mathematical dependencies where transitions reference places outside arc-connected locality. A transition t can sense M(p_remote) without any arc connecting them. This enables distance-independent regulation through kinetic equations where places p_remote may be topologically distant. Remote sensing operates through equation-implicit references without token transfer. Critically, the remote-sensed place p_remote need not belong to , need not carry any arc, and need not participate in the commitment path in any way. A Layer-3 execution sigma factor can sense a Layer-0 metabolite via (feedback inhibition); a metabolic rate can depend on a downstream product (end-product inhibition); any place in P may appear in any rate function. Because creates no arc, none of these dependencies appear in and none can create a preemption cycle. The acyclicity constraint governs only topology; is completely unrestricted in its read set. Table 1 summarises the key differences between the two access modes.

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Table 1. Connected vs. Remote Information Propagation.

https://doi.org/10.1371/journal.pone.0355540.t001

Remark ( and serve structurally distinct roles): Rate functions are inherited unchanged from standard Bio-PN. This follows the two-layer architecture of stochastic Petri nets [13]: the structural enabling layer (Enabled predicate, topology-based) determines whether a transition may fire; the kinetic propensity layer () determines how fast it fires. therefore does not appear in the Enabled predicate or the Firing rule by design—this is the standard separation in stochastic PN, not a deficiency. governs simulation dynamics (propensity for -leaping, ODE rates). A natural question is whether , being able to encode any function of any state variable, makes redundant. It does not: operates over —modelling continuous, reversible kinetics whose read dependencies may be cyclic—while operates over —modelling discrete, irreversible commitment whose topology must be acyclic. The structural theorems (Acyclicity, Preemption, Basin Partition) are properties of : they cannot be recovered from rate functions alone because is not subject to the DAG constraint that makes them provable. In the limit—a net with all regulation in and no arcs—the marking never changes from M₀ and there is no basin boundary. The moment you add a signal flow arc you get an irreversible commitment boundary with threshold . expresses what a transition reads; only arcs determine what it destroys, and it is destruction that makes commitment irreversible.

3.3. Hierarchical enablement

Definition 4 (Hierarchical Enablement Predicate). Transition t is hierarchically enabled at marking M if:

(1)

where we define notation:

• (normal input places)
• (normal output places)
• (signal input places)
• (signal output places)
• (transitions producing place via signal arcs)
• (transitions consuming place via signal arcs)
• (test arc places)

and predicates:

(2)

(3)

(4)

(5)

The PreemptionCheck enforces hierarchical dependencies by requiring all transitions producing input signal places (via signal flow arcs) satisfy their local enablement conditions. This is a single-layer check: only immediate signal-producing predecessors are verified, not transitive ancestors. Hierarchical consistency propagates automatically because each layer’s PreemptionCheck verifies its direct predecessors, creating cascading verification back to Layer 0 through the acyclic structure.

3.4. Firing rule

Definition 5 (Firing Function). Upon firing enabled transition t at marking M, the firing function Fire(t,M) produces new marking where:

(6)

with token changes:

(7)

(8)

Test arcs do not affect marking: implies (non-consuming observation).

Remark (sensing threshold vs. kinetic weight): The predicate TestEnabled uses the sensing threshold , which is distinct from the kinetic weight appearing in the rate function . encodes the activation level of the local wired sensor—the minimum signal required to declare the transition enabled—while is a kinetic modulation parameter (e.g., in a Michaelis-Menten term). These quantities have different scaling laws: when initial conditions are rescaled by factor , explicitly-set must scale by to preserve enablement semantics, while (an intensive parameter embedded in ) is unchanged. The default implements a pure presence check (M(p) > 0), using the strict inequality in the TestEnabled predicate, which is scale-invariant and biologically appropriate when catalytic action requires only that the species be present.

Dual arc case: When with both and , firing consumes tokens from both normal and signal weights: .

Signal consumption depletes regulatory tokens, creating state space partitions: markings below commitment threshold cannot reach markings above after firing.

Algorithm 1 Hierarchical Layer Assignment

Require: Signal flow graph

Ensure: Layer function or FAIL if contains cycle

1: for all

2: for all      ▷ Count incoming signal paths

3:                ▷ Source signal places

4:

5: while do

6:  

7: 

8:  for each and do     ▷ Outgoing signal paths through t

9:   

10:  

11:   if then

12:     

13:   end if

14:  end for

15: end while

16: if then

17:  return FAIL               ▷ Cycle detected

18: else

19:  return

20: end if

3.5. Layer assignment algorithm

Complexity: via modified topological sort. Algorithm fails if contains cycles, providing constructive verification of acyclicity theorem.

3.6. Commitment Semantics

Commitment occurs when transition fires from basin boundary marking:

1. Pre-commitment (before firing): — signal not yet at threshold, transition disabled, cell uncommitted
2. Commitment boundary: — transition enabled; firing creates irreversibility
3. Post-commitment (after firing): — signal tokens consumed, residual marking prevents return; cell committed

The key insight: signal consumption creates one-way transition. Unlike normal Petri nets where reverse firing can restore markings, signal depletion establishes state space partitioning—committed cells cannot uncommit through any firing sequence. Once a cell commits to sporulation by firing from basin boundary, consuming signal tokens, there is no alternative biochemical path back to the vegetative state through the same connected hierarchy. The commitment becomes final—the cell must follow the sporulation program forward. This captures biological reality: developmental commitment proceeds irreversibly through forward-only paths, even though metabolic feedback loops maintain homeostasis through remote sensing mechanisms.

The three-phase commitment semantics are directly observable in these trajectories (Fig 2). Before all 50 replicates are in the pre-commitment phase () and all downstream layers are structurally disabled by PreemptionCheck regardless of Spo0AP abundance. At the stochastic ramp crosses in a subset of replicates: these enter the commitment boundary and fire , consuming GTP tokens via the signal-flow arc irreversibly. Post-commitment, all downstream layers (, , , , forespore assembly) fire in near-deterministic sequence — the red fan in the top strip is visibly tighter than the phosphorelay fan below, quantifying the noise suppression imposed by hierarchical gating. Replicates that never cross (grey) remain in the vegetative basin with indefinitely: no alternative firing sequence can reach the sporulation attractor.

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Fig 2. PreemptionCheck in action: stochastic commitment collapses to quasi-continuous execution.

Individual trajectories of three key cascade species across 50 stochastic -leaping replicates of the B. subtilis sporulation model (M, SinR M, run_20260719_170034). Red: sporulating replicates (24/50); grey: vegetative replicates (26/50). Thick lines: fate-conditioned means. Bottom — Spo0AP: stochastic phosphorelay; both fates share indistinguishable variance until . Middle — : commitment gate; violet dashed line marks M; only sporulating replicates cross it. Top — : execution programme; silent in vegetative replicates; activates post-commitment with markedly reduced variance, demonstrating the quasi-continuous character of execution once PreemptionCheck unlocks. The effective threshold (M, identified via simulation) separates sporulating from vegetative replicates: once crossed, GTP tokens are consumed irreversibly via the signal-flow arc and all downstream layers fire in lock-step.

https://doi.org/10.1371/journal.pone.0355540.g002

4.1. Formal definition

Definition 6 (Signal Hierarchical Petri Net). A Signal Hierarchical Petri Net is a 13-tuple:

(9)

where:

• (P, T, F, W, M₀) is classical 5-tuple (places, transitions, arcs, weights, initial marking)
• assigns kinetic rate laws to transitions. governs the simulation dynamics (propensity for -leaping, ODE rates). Following the standard two-layer architecture of stochastic Petri nets [13], does not appear in the structural Enabled predicate: enabling is topological, propensity is kinetic.
• assigns each place to a spatial compartment (default: cytoplasm), used by the simulation engine for volume-based concentration conversion () and multi-compartment diffusion. C does not appear in the structural Enabled predicate or Firing rule: compartment assignment is a simulation-level parameter, not a topological one.
• defines test arcs (catalysis, non-consuming); test arc enablement requires with default (presence check M(p) > 0) — the sensing threshold is uniformly 0 unless explicitly set
• defines signal places (information channels)
• defines signal flow arcs
• assigns signal arc weights
• assigns hierarchical layers
• assigns enablement thresholds

4.2. Architectural principles

Vertical-Horizontal Separation: Normal arcs F model horizontal mass transfer (metabolic reactions), signal flow arcs model vertical information propagation (regulatory control). Signal places function as information hubs: metabolic transitions connect via normal arcs to modify at Layer 0 (horizontal participation), while regulatory transitions connect via signal flow arcs to consume signals and broadcast state changes to higher layers (vertical participation). This architectural separation mirrors biological organization where metabolic networks operate horizontally through spatially-distributed enzyme cascades, while regulatory hierarchies operate vertically with metabolic state gating developmental decisions through connected signal paths.

Connected vs. Remote Information Transfer: The formalism provides two complementary mechanisms. Connected channels (signal flow arcs ) implement topology-explicit wired propagation where physical arc connections enable hierarchical preemption—consumption at layer k triggers re-evaluation at all layers . Remote sensing (rate functions ) implements equation-implicit wireless observation where transitions reference concentrations of topologically distant places without physical connection—enabling long-range regulation through kinetic dependencies without hierarchical influence. Biologically, connected channels model committed resource allocation with hierarchical consequences ( depletion below M at commitment disables all downstream execution layers), while remote sensing models long-range regulatory effects without physical transport (Spo0AP concentration influencing distant operon expression through equation dependencies).

Dual Arc Semantics: Signal places can simultaneously participate in both normal arcs (horizontal mass transfer) and signal flow arcs (vertical information propagation):

(10)

(11)

When transition t fires with both arc types from place , marking evolution is:

(12)

This enables the same metabolite (e.g., ATP) to participate in biochemical reactions (glycolysis) and regulatory decisions (commitment gating) without duplication.

Design principle — selective wiretap on the physical layer: Signal type assignment does not introduce new arcs, new places, or new tokens. It is a selective annotation: the modeler designates certain existing normal arcs as signal flow arcs, so that the same token-consuming event that performs biochemistry in simultaneously reports a commitment to the control hierarchy . The cell does not maintain two separate copies of a molecule—one for metabolism, one for signaling. There is one token; the type annotation gives its consumption a second, hierarchical meaning. This is why the two-graph architecture is necessary: must be free to cycle (metabolic feedback, oscillations, reversible reactions), while must be acyclic (commitments are one-way). The same token stream feeds both graphs; the arc type determines which graph records the event.

4.3. Operational semantics

Definition 7 (Marking Evolution). State space consists of all possible token distributions. Single-step reachability relation is:

(13)

Multi-step reachability is the reflexive transitive closure: holds if there exists firing sequence with for some .

Definition 8 (Reachability Set). The reachability set from initial marking M₀ is:

(14)

Signal consumption creates reachability partition: where basin boundaries form one-way barriers (Theorem: Irreversible Basin Partition).

Proposition 1 (Hierarchical Decomposition). For signal hierarchical PN with k layers, reachability analysis decomposes into layer-wise subproblems. Analyzing Layer with fixed commitments from layers reduces complexity from (product of layer state spaces) to where is branching factor at layer i.

Proof. General reachability in Petri nets is Ackermann-complete [19] (and hence not primitive-recursive), with state space that grows non-primitively-recursively in the number of places. For flat model with places distributed across k layers, reachability graph has states where C is token bound.

Hierarchical structure enables decomposition: (1) Analyze Layer 0 independently, computing commitment thresholds via (O(1) per transition). (2) Fix Layer 0 commitments, reducing Layer 1 to subproblem with Layer 0 constraints—state space now where b₀ bounds Layer 0 commitment states. (3) Repeat for layers .

Total complexity: sum over layers instead of product. For B. subtilis model (40 places, 4 layers): flat analysis requires O(10⁴⁰) states; hierarchical analysis requires states—a reduction of 30 orders of magnitude that enables practical computation. □

4.4. Comparison with classical Bio-PN

SHPN extends classical Bio-PN expressiveness through signal-type semantics, as summarised in Table 2.

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Table 2. Expressiveness Comparison: Classical Bio-PN vs. Signal Hierarchical PN.

https://doi.org/10.1371/journal.pone.0355540.t002

Key distinction: Test arcs enable transitions when without token consumption, maintaining after firing. Signal flow arcs require and consume tokens, producing after firing. This semantic difference between consumption and read-only behavior determines expressiveness: test arcs model catalysis such as enzyme-substrate binding, while signal flow arcs model commitment through decision quota depletion.

5.1. Signal place designation

The formalism’s power derives from modeler choice: any metabolite can be designated as signal place when biological evidence supports regulatory gating role. This designation determines dual participation through horizontal mass transfer and vertical information broadcast.

Horizontal mass transfer. Signal places are normal places participating in biochemical reactions via normal arcs F at Layer 0. For example, ATP in glycolysis: with W = 2 (stoichiometric production). NADH in oxidative phosphorylation: with W = 1 (consumption).

Vertical information broadcast. The same places, when we assign type “signal,” broadcast availability to higher regulatory layers via signal flow arcs . For example, GTP_pool gating septation: with (structural commitment arc consumed on firing). Spo0AP broadcasting phosphorelay output: (integration-to-commitment gating). Note that gates T_septation kinetically via rate function (remote sensing), not via a signal flow arc—the commitment separatrix M is a kinetic threshold identified by simulation, not a structural value.

This type assignment creates an interface from Layer 0 to higher layers: signal places and signal flow arcs are ordinary places and arcs to which we assign signal semantics, enabling the same places to participate in horizontal mass flow (metabolic transformations) and vertical information broadcast (regulatory propagation). Mass flows horizontally through stoichiometric reactions; information about mass availability broadcasts spatially (vertically) to the cellular regulatory environment. No artificial separation into distinct species—unified place, differentiated semantics through arc type assignment.

Biological case selection: For B. subtilis sporulation, we designate as the commitment-gate signal place and Spo0AP as the phosphorelay integration signal [14]. ATP and GTP remain metabolic participants (via normal arcs) but are not the separatrix species: the commitment threshold is the separatrix (M), not an energy quota. Other systems designate different signals: calcium (Ca²⁺) for neurotransmitter release, cyclic AMP (cAMP) for bacterial chemotaxis, GTP for protein synthesis quality control, NADH for metabolic oscillations. The formalism generalizes across signal types—designation derives from biological function, not molecular identity.

5.2. Four-layer architecture with multi-level signal assignment

Signal places can be designated at any hierarchical layer, not restricted to Layer 0. This design choice significantly enhances formalism expressiveness by enabling multi-stage commitment with cascading irreversibility. For the B. subtilis case, signal places are designated across all four layers as determined by the signal flow arc topology of the v9 model:

Layer 0 (Starvation Sensing): KinA_kinase designated as signal (). Rate : active only under starvation. Signal flow arcs connect KinA_kinase to the phosphorelay entry, creating the first commitment barrier: absence of nutrient stress structurally disables all downstream layers regardless of phosphorylation state.

Layer 1 (Phosphorelay Integration): Spo0AP designated as signal (). Signal flow arcs carry the phosphoryl group through the four-protein relay (KinA_kinase KinA_P Spo0F_P Spo0AP), each step stochastic. Spo0AP broadcasts to Layer 2, creating the second commitment barrier: insufficient phosphorelay output prevents commitment-gate activation.

Layer 2 (Commitment Gate): designated as signal (). Produced by a Hill-4 positive-feedback switch on Spo0AP (: ); gates T_septation kinetically via remote sensing (). The effective threshold M is identified via simulation (Sweep A): only replicates crossing it fire T_septation, which consumes GTP tokens via the structural signal-flow arc, unlocking Layer 3.

Layer 3 (Execution): , , , designated as signals (). Continuous sigma factor cascade activated post-septation. Signal flow arcs from Layer 2 unlock transcription; once active, execution proceeds with markedly reduced variance (quasi-continuous regime, visible in Fig 2): sigma factor depletion can no longer reverse the committed phenotype.

This multi-level signal architecture enables cascading irreversibility: each layer establishes its own basin boundaries through signal consumption. A cell can be starvation-committed (Layer 0 active) yet phosphorelay-uncommitted (Layer 1 below threshold), creating intermediate commitment states impossible if signals were restricted to a single layer. The formalism thus captures multi-stage biological decision-making with progressive commitment refinement.

5.3. Information flow topology

Signal flow graph encodes regulatory architecture through connected channels created by type assignment. Signal places function as hierarchical information hubs at each layer, not restricted to Layer 0. For the B. subtilis case, we designate spanning all four layers. Layer assignment , , , creates the multi-level signal architecture.

Connected paths define hierarchical preemption cascades. For example:

• : Layer 0 starvation gating (signal flow arc)
• : Layer 01 phosphorelay broadcast (signal flow chain)
• : Layer 12 Hill-switch commitment gate (rate function)
• : Layer 23 execution unlock (rate function remote sensing)

Each signal place at layer k can be produced/consumed via normal arcs (horizontal mass transfer within layer) and broadcast via signal flow arcs to layer k + 1 (vertical information propagation across layers). This architecture enables cascading commitment: depletion at any layer preempts all higher layers through hierarchical preemption, while each layer maintains its own basin boundaries.

Acyclicity enforced: no connected path exists through signal flow arcs—execution sigma factors cannot re-activate the starvation sensor through hierarchical commitment channels. Remote sensing through rate functions enables equation-implicit feedback without creating preemption cycles: transcription can sense any place—including upstream metabolites, downstream products, or places with no arc at all—via without physical arc connection. The remote site is not required to be in the commitment path, not required to be a member of , and not required to have any arc. Feedback therefore remains fully unrestricted (metabolic oscillations, product inhibition, allosteric regulation) while the irreversible commitment structure stays acyclic. The formalism thus captures both hierarchical stratification (acyclic ) and full feedback biology (cyclic remote dependencies in ) independently.

5.4. Threshold computation

The formalism enables topology-driven threshold computation. For any designated signal controlling transition t, the commitment marking follows:

(15)

This formula expresses commitment finality: firing from M_commit consumes decision quota , leaving residual exactly—the minimal marking maintaining enablement. Below M_commit, firing disabled. Above, firing possible but optional. At boundary, firing creates irreversibility through basin partitioning.

The formula generalizes across signal types: calcium-triggered neurotransmitter release designates Ca²⁺, cAMP-triggered chemotaxis designates cAMP, protein synthesis quality control designates GTP. Threshold computation depends on topology and biological designation, not molecular identity. The B. subtilis sporulation case demonstrates this principle with as the designated commitment-gate signal (M) and Spo0AP as the phosphorelay integration signal.

6.1. Mathematical validation

6.1.2. Theorem: Preemption correctness.

Hierarchical enablement predicate correctly implements preemption semantics.

Proposition 3 (Hierarchical Consistency). If Enabled(t, M) = true under hierarchical predicate, then all lower-layer signal place producers satisfy their enablement thresholds.

Proof. We prove by strong induction on layer depth where is the maximum layer of signal inputs to t.

Base case (): Transition t has no signal inputs from higher layers, only Layer 0 metabolic places. PreemptionCheck(t,M) vacuously true (no lower-layer predecessors to check). SignalEnabled(t,M) requires only Layer 0 thresholds, which are satisfied by definition of Enabled(t,M).

Inductive step: Assume for all transitions with , if Enabled then all their lower-layer predecessors satisfy thresholds. Consider transition t with .

By definition of Enabled(t,M), we have PreemptionCheck(t,M) = true, which requires:

For each signal input with , all producing transitions satisfy their local enablement conditions. Since , these transitions have . By inductive hypothesis applied to each , all their direct signal predecessors satisfy thresholds.

Hierarchical consistency propagates through the acyclic structure: verifying immediate predecessors at each layer creates cascading verification along signal flow paths . Each transition ’s PreemptionCheck verifies , creating chain of verification back to Layer 0 sources without requiring explicit transitive checking. □

This guarantees ATP depletion cascades upward: if Layer 0 fails threshold, all dependent layers disabled regardless of local token abundances.

6.1.3. Corollary: Threshold predictability.

Basin boundaries are structural properties computable without simulation.

Proposition 4 (Computational Threshold Prediction). For transition t with signal input , commitment threshold is computable in O(1) time from formalism specification without state space exploration.

Proof. By Definition (Basin Boundary), commitment marking is defined as . Both components are formalism parameters:

• is specified via threshold function (part of 13-tuple definition)
• is specified via signal weight function (part of 13-tuple definition)

Computation procedure:

1. Identify signal input arc: (graph traversal, worst case)
2. Retrieve threshold: (function evaluation, O(1))
3. Retrieve weight: (function evaluation, O(1))
4. Compute commitment: (addition, O(1))

No simulation, no reachability analysis, no numerical integration—direct arithmetic from model specification. This contrasts with the Ackermann-complete reachability analysis [19] required for general threshold determination in unstructured Petri nets. □

This establishes topology-driven predictability: the threshold is structurally localized — once the signal topology is fixed, it is determined by at most two measurable quantities per signal arc ( and ), extracted from biological literature (BRENDA database, biochemistry textbooks). The claim is not that the threshold is parameter-free, but that it does not require fitting a global nonlinear model: extract parameters from biological literature, instantiate formalism functions and , compute threshold via closed-form formula, obtain quantitative prediction without simulation or parameter optimization. For the B. subtilis sporulation model this yields the effective commitment threshold, identified via simulation in Proposition 5 in Section 6.

Remark (Kinetic constraint tuple and effective basin floor ): Each signal flow arc carries a kinetic constraint tuple with three components drawn from published enzyme kinetics: is the Michaelis constant (same units as marking, e.g., M), n > 0 is the Hill coefficient (cooperativity exponent), and is the suppression floor fraction. The effective basin floor is computed analytically from :

(16)

When (default), and enablement reduces to . The threshold component of the 13-tuple evaluates to for each signal input arc .

The claim that is “not fitted” operates at two distinct levels. (1) Cross-domain grounding: is sourced from molecular kinetics data (BRENDA, published Hill-equation fits) — a different experimental domain from the sporulation fraction being predicted. These parameters describe enzyme–substrate binding behaviour; they were not adjusted to match any cellular-level observation. (2) Analytical derivation: and are computed once by formula, with no tuning step targeting the Fujita fractions. The validation claim is therefore that molecular-scale constants, sourced independently, predict a cell-population-scale commitment threshold without any intermediate fitting targeting the predicted outcome.

6.2 Biological validation: Bacillus subtilis sporulation

6.2.2. Model construction.

The model instantiates the four-layer signal hierarchical architecture (Section 5.2). Model statistics: 40 places, 36 transitions, 114 arcs, 4 signal layers (Fig 3).

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Fig 3. Signal Hierarchical Petri Net: B. subtilis Sporulation (v9).

Full topology of the B. subtilis sporulation Signal Hierarchical Petri Net. Circles (•) are biological places; hexagons () are signal places; diamonds () are spatial-signal places; rounded squares () are parameter places. Black solid lines: normal arcs (stoichiometric consumption); light-grey dashed lines: signal-flow arcs; blue dashed lines: test arcs (non-consuming, presence check only); green rectangles: source/sink transitions. The four signal layers are: Layer 0 (KinA_kinase, starvation sensing), Layer 1 (Spo0AP, phosphorelay integration), Layer 2 (, commitment gate, separatrix M), Layer 3 (, , , , execution programme). Parameter places (top row) encode the experimental protocol (nutrient level, temperature, loading dose) and are read by events only—they carry no arcs and do not appear in any rate function .

https://doi.org/10.1371/journal.pone.0355540.g003

6.2.6. Basin of attraction analysis.

The landscape (Fig 4) reveals three dynamical regimes along the time axis. Before the landscape is a single vegetative well (): all 50 replicates are trapped at low order parameter regardless of stochastic fluctuations, because and remains blocked. Around the barrier lifts (coefficients b(t) and c(t) transition via ), the ridge emerges, and the mean trajectory begins to rise. After a sporulation attractor forms in the upper region (), separating from the persistent vegetative well at . The near-symmetric double-well at late times encodes the sporulation fraction: neither basin is strongly favoured, reflecting the bet-hedging character of the commitment decision.

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Fig 4. Waddington epigenetic potential landscape of B. subtilis sporulation commitment.

Analytic quasi-potential () with time-varying coefficients calibrated to the commitment clock (, median crossing at M). Dark blue = low-energy attractor basins; dark red = energy barrier (Waddington ridge). White line: mean sporulation order parameter averaged over 50 stochastic -leaping replicates (run_20260719_170034, M, 6-h horizon). Yellow circle (•): commitment point at ; green star (⋆): first appearance of mature spores (). Red dashed line: separatrix. The landscape reproduces the bet-hedging outcome reported by Fujita & Losick [14]: sporulation under nutrient-rich conditions (M, SinR M, no drug pulse).

https://doi.org/10.1371/journal.pone.0355540.g004

Corollary 5 (Basin Boundary Characterization — B. subtilis). For the B. subtilis sporulation model, the reachability space partitions into two regions. Using the notation of Theorem (Irreversible Basin Partition) with the SinR/ commitment gate:

where GTP_pool is the signal-flow arc input to T_septation and . The SinR inhibitor arc (threshold M) and the kinetic factor constitute the complementary structural and kinetic commitment barriers. No firing sequence from B_below reaches B_above after T_septation fires.

Proof. Direct application of Theorem (Irreversible Basin Partition) with GTP_pool as the signal-flow arc input to T_septation. When the SinR/SinI gate opens and is above the effective kinetic threshold, T_septation fires from B_boundary, consuming GTP tokens via the signal-flow arc and producing Septum via the curved signal-flow arc. The resulting Septum inhibitor arc (, curved inhibitor) then blocks re-firing, making the partition absorbing. With no arc regenerating GTP_pool after consumption, B_below is absorbing per the theorem. □

6.3. Developmental scope and pre-print record

During the development of signal hierarchy theory, the formalism was applied to two further biological systems. A predecessor analysis of lambda phage lysis-lysogeny, using an information-theoretic approach that preceded the full SHPN arc-type semantics, is available as a pre-print [23] (arXiv:2512.22415, submitted December 2025). An earlier 13-tuple formulation applied to V. fischeri quorum sensing is available as a pre-print [24] (arXiv:2601.00036, posted January 2026). Neither is presented as SHPN validation here: both pre-dates the signal-consumption semantics formalized in this paper, and both are being revised under the mature definition before formal journal submission. A GATA1/PU.1 haematopoietic lineage commitment model applying the same formalism is in active preparation (E.S., unpublished). These developmental works establish breadth of applicability and are noted here to avoid duplication should readers encounter the pre-prints independently.

8.1. Theoretical contributions

Signal hierarchy theory advances Petri net foundations by proving two structural properties distinguishing hierarchical commitment from general regulation. The acyclicity theorem formalizes hierarchical stratification for connected commitment channels: metabolism precedes sensing precedes integration precedes execution through signal flow arcs, forbidding circular preemption dependencies where committed decisions create their own enabling conditions. Importantly, this constraint applies only to topology-explicit connected structure ( arcs defining preemption), not to equation-implicit remote sensing ( enabling feedback through kinetic dependencies). The preemption theorem establishes vertical dependencies: lower-layer threshold failures propagate upward through connected channels, disabling higher layers regardless of local token abundances. These theorems codify the distinction between hierarchical commitment (must be stratified, acyclic connected paths) and feedback regulation (can be cyclic, remote sensing without preemption coupling), enabling formal verification of hierarchical correctness while preserving regulatory feedback capabilities.

Consumptive information semantics differentiate signal flow arcs from test arcs, resolving expressiveness limitations of classical Bio-PN. Test arcs model catalysis (enzymes reading substrate concentrations without consumption), signal flow arcs model commitment (decision systems depleting regulatory quotas during irreversible choices). This semantic distinction—read-only vs. consumptive—determines predictive capability: test-arc models require experimental threshold measurements, signal-flow models compute thresholds from topology. The innovation: information can be consumed like mass but transferred like signals, creating hybrid entity (regulatory tokens) enabling threshold computability.

Basin boundary computability establishes thresholds as structural properties. Classical dynamical systems require simulation to find attractor basins—computationally expensive state space exploration. Signal hierarchical networks provide closed-form formula: . No simulation, no fitting, no iteration—extract parameters from literature, apply formula, obtain quantitative prediction. This reduces threshold determination from Ackermann-complete reachability problem [19] to O(1) arithmetic, enabling genome-scale analyses infeasible with simulation-based methods.

Relationship to related formalisms: Beyond classical Bio-PN, SHPN relates to a broader tradition of process-oriented and partial-order-based approaches. Condition/Event nets (C/E nets [28]) distinguish conditions (Boolean state) from events (occurrences), encoding the non-consuming vs. consuming distinction at the token level—the closest precursor to the test-arc / signal-arc distinction in SHPN. Occurrence nets and net unfoldings [26,27] establish acyclicity in the causal execution record, capturing the irreversibility of events in the temporal unfolding of concurrent processes. Process-algebraic approaches (Regev and Shapiro [29], Priami [30]) model individual molecular interactions concurrently through process composition, excelling at combinatorial signaling and competitive binding. These traditions address complementary aspects: process algebras model concurrent interaction identity; occurrence nets model causal irreversibility in execution history; SHPN models population-level commitment thresholds through a structural constraint on the control topology itself. In Petri’s partial order tradition, acyclicity is a property of the execution record (the unfolding is always a DAG even if the underlying net has cycles). SHPN adds a second, independent acyclicity requirement: must be a DAG as a condition on the net structure—because biological commitment channels are irreversible by design, not merely by the history of one particular run.

8.2. Methodological advances

Unified metabolic-regulatory modeling addresses the architectural separation of metabolic and regulatory concerns in classical systems biology approaches. FBA optimizes metabolism ignoring regulation. Boolean networks model regulation treating metabolites as parameters. Hybrid models separate concerns with artificial interfaces. Signal hierarchical PN unifies through modeler designation: any metabolite becomes signal place when biological function supports it, participating in biochemical networks (normal arcs, horizontal mass transfer) and regulatory hierarchies (signal flow arcs, vertical information propagation) simultaneously. No duplication, no parameterization, no separation—integrated species with differentiated arc semantics. For B. subtilis, we designate as the commitment-gate signal and Spo0AP as the phosphorelay integration signal; for calcium signaling, Ca²⁺ is designated; for chemotaxis, cAMP. This architectural unification enables analyses impossible in fragmented frameworks: How does metabolic perturbation propagate through sensing layers to alter commitment thresholds? Unified model answers directly; fragmented models require manual integration.

Topology-driven threshold computation distinguishes this work from parameter-fitting approaches. Classical workflow: build model, measure thresholds experimentally, fit parameters, validate. Signal hierarchical workflow: designate signals from biological evidence, extract arc parameters from published biochemical constants, compute , compare to independent experimental data. The structural inversion: the threshold is an output of the topology, not an input to a fit. For B. subtilis sporulation, the separatrix at M accounts for the Fujita and Losick outcome asymmetry; the arc parameters were drawn from independent biochemical data, not tuned to that ratio. This distinction matters for interpretation: a correctly parameterized mechanistic model is expected to reproduce experimental observations, because correct parameters reflect real molecular properties. The match is therefore not coincidental — it is evidence that the parameterization is correct. What makes it a genuine validation rather than a tautology is that the Fujita ratio was never consulted during model construction: the 52%/5% asymmetry could not have been obtained by other means without also capturing the correct separatrix and phosphorelay dynamics.

Hierarchical decomposition ameliorates computational complexity. Reachability in Petri nets is Ackermann-complete [19], limiting classical methods to small models (<50 places). Signal hierarchy enables divide-and-conquer: analyze Layer 0 independently (metabolic production of designated signals), compute commitment thresholds, use thresholds as Layer 1 constraints, analyze sensing dynamics, repeat upward. The complexity reduces from exponential product over layers to linear sum times branching factors. B. subtilis model (40 places, 4 layers) analyzed in 2.3 seconds; equivalent flat model requires estimated 10 + hours. Hierarchical structure converts intractable problem into practical computation.

8.3. Biological insights

Signal designation as gatekeeper: Modeler-designated signals create biological firewalls preventing infeasible commitments. For B. subtilis, is designated as the commitment-gate signal (Layer 2). Commitment without crossing the separatrix (M) is structurally impossible: the Preemption theorem disables all downstream execution layers (, , , ) regardless of Spo0AP abundance. Biology implements a fail-safe: the commitment gate vetoes execution when the gating signal is insufficient, preventing premature or incomplete developmental programs. This principle generalizes: Ca²⁺ depletion disables neurotransmitter release, GTP depletion halts protein synthesis, cAMP depletion blocks chemotaxis—all through the same hierarchical preemption mechanism applied to different designated signals.

Irreversibility through consumption: Commitment creates one-way transitions via signal depletion. Sporulation initiation consumes regulatory tokens (activated Spo0AP), establishing basin boundary: pre-commitment cells (below threshold) occupy reversible attractor basin; post-commitment cells (above threshold, fired) occupy irreversible basin. No firing sequence reverses—signal consumption prevents return. This explains experimental observations: committed cells complete sporulation even when nutrients return (irreversibility); uncommitted cells return to vegetative growth when nutrients restored (reversibility). Mathematical model captures biological finality through consumptive semantics.

Hierarchical signal integration: Multi-layer architecture enables cascading irreversibility (Section 5.2): each layer establishes its own basin boundary via signal consumption, and depletion at any layer propagates upward through PreemptionCheck to disable all higher layers. This stratification enables robust decisions: single sensor failure does not prevent commitment (redundancy), but designated signal collapse does (preemption). Hierarchy balances reliability (multiple pathways) with safety (designated resource gating). The architecture generalizes: calcium signaling designates Ca²⁺, chemotaxis designates cAMP, transcription quality control designates GTP—same hierarchical organization, different signal identities.

8.4. Limitations and future directions

Spatial heterogeneity: Current formalism assumes well-mixed compartments. Bacterial sporulation exhibits spatial asymmetry: mother and forespore compartments differ in regulatory states. Extending signal hierarchical PN with compartmentalization (colored tokens, compartment-specific transitions) would enable spatial commitment modeling. Challenge: maintaining threshold computability when diffusion creates continuous spatial gradients.

Stochastic dynamics and noise analysis: The hybrid formalism supports both deterministic and stochastic simulation engines, including tau-leaping and Stochastic Simulation Algorithm (SSA) implementations enabling efficient stochastic parallelism. For the B. subtilis case study, deterministic kinetics suffice for threshold prediction because commitment thresholds represent mean-field basin boundaries—population-level decision points independent of single-cell trajectory noise. However, stochastic engines become essential when analyzing commitment reliability under low-copy-number fluctuations (e.g., transcription factor noise affecting decision timing) or computing commitment probability distributions. The challenge: extending topology-based threshold formulas to stochastic settings where basin boundaries become probability distributions rather than deterministic values. Future work includes developing analytical approximations for commitment probability as function of signal variance.

Parameter uncertainty: Literature parameters have experimental error (-50%). Current sensitivity analysis shows robustness to individual parameter variations, but systematic uncertainties require probabilistic treatment. Bayesian parameter estimation over signal hierarchical models would quantify prediction confidence intervals. Challenge: maintaining topology-driven philosophy when incorporating posterior distributions from data.

Genome-scale models: B. subtilis model has 40 places; genome-scale metabolic models exceed 1000 metabolites. Hierarchical decomposition enables scaling, but manual layer assignment becomes infeasible. Developing automated layer inference algorithms (e.g., graph partitioning optimizing vertical-horizontal separation) would enable genome-scale signal hierarchical models. Challenge: biological hierarchy not always obvious from network topology alone.

Experimental validation: Current validation compares predicted thresholds to published measurements. Stronger validation: experimentally perturb predicted parameters (e.g., engineer ATP synthase mutants), predict shifted thresholds, measure in vivo. Prospective predictions (before measurement) provide more rigorous tests than retrospective comparisons (after measurement).

8.5. Applications

Synthetic biology: Designing decision circuits with predictable commitment thresholds requires the threshold to be computable from the circuit topology before construction. Current practice: build circuit, measure threshold, iterate. Signal hierarchical approach: specify desired threshold, compute required signal weights backward (), construct circuit matching specifications.

Drug discovery: ATP-dependent pathways represent therapeutic targets. Cancer cells commit to proliferation at higher ATP thresholds than normal cells, creating therapeutic window. Signal hierarchical models predict: (1) which pathways exhibit ATP-dependent thresholds, (2) threshold values for normal vs. cancer cells, (3) compounds shifting thresholds (ATP synthase inhibitors). Enables prioritizing targets with maximum differential between cell types.

Patient-specific parameterisation: Signal hierarchical models can, in principle, be instantiated from patient-specific enzyme and metabolite measurements to compute individualised commitment thresholds. This direction requires experimental validation of the topology-first assumption across heterogeneous populations and is reserved as future work.

Systems biology: Understanding cellular decision-making requires integrated metabolic-regulatory models. Signal hierarchical PN enables: (1) computing how metabolic perturbations (nutrient shifts) alter commitment thresholds, (2) analyzing commitment reliability under energetic fluctuations, (3) designing experiments probing basin boundaries. Broader impact: framework applicable beyond bacteria—mammalian cell cycle, immune cell differentiation, stem cell fate decisions all exhibit energy-gated commitment with predictable thresholds.