A Nonlinear Model of Complementary Regulatory Dynamics (CRD): Implications for Systems Biology and Dynamical Systems Medicine

1. Introduction: From Functional Architecture to Dynamical Model

Living systems must continuously reconcile two seemingly conflicting requirements: responsiveness for adaptation while maintaining stability. For example, cells must activate and terminate signaling, and immune systems must mount protective responses and resolve inflammation [1,2]. Established concepts address some of these problems, including homeostasis, allostasis, robustness, feedback regulation, adaptation, and resilience [3,4,5,6]. Dynamical systems theory may also explain how interactions among system components generate stable states, oscillations, transitions, and pathological regimes [7,8]. However, across these fields, a practical need remains for unifying, biologically meaningful models that are abstract enough to analyze, compare, and extend, yet still support detailed explanations.

Complementary Regulatory Dynamics (CRD) was introduced as an integrative theory to address this need and help meet the challenges in systems biology [9]. CRD may provide a common logic for understanding Complex Adaptive Systems (CAS) and help establish causal bridges across multiple scales. It proposes that sustained, system-relevant change can often be analyzed in terms of two types of processes with outcome-relative functional roles. One is a functional Activator role, which initiates, promotes, amplifies, or propagates. The other is a functional Regulator role, which constrains, inhibits, buffers, restores, redirects, or terminates. Together, these roles form an Activator-Regulator (A-R) pair, and their interactions drive change [9].

The identities of the relevant processes depend on the stated system boundary, outcome, organizational level, condition, and time scale [9]. Thus, an Activator or Regulator is not an intrinsic property of a molecule, cell type, organ, or behavior. The same component may assume different functional roles under different conditions or at different scales. CRD does not propose that all systems comprise two fixed entities, because the roles are functional, relative, and context-dependent [9].

CRD theory has five provisional, testable principles: functional complementarity, mutual regulation, dynamic balance, transformation, and multiscale recurrence (functional fractality) [9]. Complementarity means that the Activator (A) and Regulator (R) have interdependent functional roles that jointly determine a shared system outcome. Mutual regulation refers to direct or indirect reciprocal regulatory interactions between these two functional types. Dynamic balance does not imply static equality between activation and regulation but rather a bounded, context-appropriate operating regime that enables responsiveness, coordination, recovery, and stability. Transformations are qualitative changes in state, coupling, role, or organization that may occur when conditions cross critical thresholds. Multiscale recurrence (functional fractality) posits that comparable functional A-R architectures recur across levels of organization [9].

These principles provide a functional vocabulary, but a scientific theory also requires models that yield quantitative predictions. Mathematical models are especially valuable because they make assumptions explicit. They require investigations to specify the processes, their interactions, the forms of nonlinearity, and which observations would support or falsify the models. Such models can also reveal when verbal intuitions fall short.

Differential equation models describe how a system’s current state influences its future state, making them suitable for studying nonlinear responses, stability, feedback, and critical transitions [7,10]. The proposed A-R architecture may not exhibit the stability, switching, or oscillation attributed to it unless specific coupling strengths, time delays, or boundary conditions are present. In this sense, formalization not only illustrates CRD but also tests and constrains it.

This paper shows how mathematical equations can formally bridge CRD’s functional architecture and dynamical systems analysis. The following sections define the variables, parameters, and interaction terms in detail and examine how the equations relate to the five CRD principles. The preliminary mathematical model may have potential applications in systems biology and dynamical systems medicine. For example, it may support representations of resilience, regulatory failure, compensatory recruitment, regime shifts, and intervention timing. As a falsifiable theory, the scientific value of the equations will depend not on their generality alone, but on whether they improve the explanation, comparison, and prediction of measurable dynamics in independently studied systems.

2. Conceptual and Mathematical Foundations of Complementary Regulatory Dynamics

2.1. The Activator-Regulator (A-R) Pair as a Functional Unit

This paper develops a minimal nonlinear model of an Activator-Regulator (A-R) pair:

$${\displaystyle \frac{dA}{dt}}=aA-bAR+hR-e{A}^3,$$

(1)

$${\displaystyle \frac{dR}{dt}}=cR-dAR+kA-g{R}^3.$$

(2)

The preliminary equations describe the time ($t$)-dependent interactions between an Activator-role variable, $A\left(t\right)$, and a Regulator-role variable, $R\left(t\right)$. The derivatives ${\displaystyle \frac{dA}{dt}} \mathrm{and} {\displaystyle \frac{dR}{dt}}$ represent their instantaneous change rates. The equations illustrate trajectories over time rather than static associations between the variables. The functional Activator role is the process that, in the stated context, initiates, amplifies, promotes, mobilizes, or propagates. The functional Regulator role is the counterbalancing process that constrains, inhibits, coordinates, buffers, restores, redirects, or terminates. These two types of functional roles are outcome-relative rather than permanent properties of molecules, cells, organs, or behaviors [9].

Thus, an A-R pair does not consist of two distinct physical entities. It can be an aggregate set of processes that jointly influence a specified outcome within a defined system boundary. The distinction between functional roles and physical components is also important mathematically. In a simple experimental system, $A\left(t\right)$ and $R\left(t\right)$ may correspond to directly measured quantities such as concentrations, cell populations, firing rates, or other physiological variables. In a more complex system, they may represent effective or aggregate variables that summarize the net action of multiple components, interactions, or processes. Section 4 provides examples of potential applications in systems biology.

For an A-R pair to be scientifically useful, it should specify at least five features. These include the system boundary, focal outcome, organizational level, relevant time scale, and the conditions under which roles are assigned. For example, a pro-inflammatory process may activate host defenses during infection, whereas pro-resolving mediators may regulate inflammation escalation and promote tissue restoration [1,2] (see also Section 4). However, the same inflammatory signal may serve a regulatory or reparative function (or “role” here) in another phase, tissue, dose range, or disease context. This contextual dependence is common in biological regulation and cautions against treating functional labels as fixed identities [11].

2.2. The Parameters and Activator-Regulator Dynamics

Table 1 summarizes the CRD model parameters. The parameters $a,\hspace{0.25em} b,\hspace{0.25em} c,\hspace{0.25em} d,\hspace{0.25em} e,\hspace{0.25em} g,\hspace{0.25em} h,\hspace{0.25em} \mathrm{and}\hspace{0.25em} k$ specify the strengths and signs of the modeled processes. In empirical applications, these are not universal constants. They may vary across tissues, organisms, developmental stages, disease states, environmental contexts, and time scales. Researchers can estimate their values from longitudinal data, infer them from perturbation experiments, or vary them systematically in simulations. This parameter dependence is essential because a regulatory relationship may be adaptive in one range of conditions and maladaptive in another [3,12].

The parameters are typically positive and real. Specifically, the terms $a$and $c$represent intrinsic growth, recruitment, maintenance, or self-reinforcement tendencies of $A$ and $R$, respectively. The interaction terms $-bAR$ and $-dAR$ represent coupling-dependent constraints, consumption, or antagonism. The cross-support terms $+hR$ and $+kA$ indicate that regulatory activity can enable productive activation, and that activation can recruit or generate regulation. The cubic terms $-e{A}^3$ and $-g{R}^3$ represent nonlinear self-limitation, saturation, resource constraints, or damping that prevents indefinite growth. Section 3 explains these terms and their roles in addressing the five CRD principles in more detail.

No single parameter determines the system’s behavior. For example, a high intrinsic Activator gain, $a$, may be offset by strong Activator self-limitation, $e$, or by a timely Regulator-dependent constraint, $b$. Similarly, a strong Activator-to-Regulator recruitment parameter, $k$, may promote recovery only if the Regulator persists long enough to act and is not excessively depleted by the $-dAR$ term. Therefore, a parameter’s interpretation depends on the full set of parameters, the initial state, and the relevant time horizon. This interdependence is central to nonlinear dynamics.

This mathematical form is deliberately minimal and simple. It preserves intrinsic activity, reciprocal coupling, cross-recruitment, and nonlinear self-limitation. The terms should be interpreted as effective descriptions of a specified system, not as literal molecular mechanisms unless those mechanisms have been independently established.

The model therefore differs from purely antagonistic activator-inhibitor or predator-prey formulations [7,10]. In Equations (1) and (2), activation and regulation can constrain one another, but under appropriate conditions they can also support one another. This distinction is important. Many living systems do not persist by simply suppressing activation. Effective regulation may allow activation to proceed within viable bounds. Appropriate regulation can also coordinate activation timing, terminate it after it has served its function, repair its consequences, or transform it into a new system state. Likewise, activation may recruit regulatory processes rather than merely oppose them. The parameters $h$ and $k$ make these complementary and potentially transformative relations explicit (see also Section 3).

3. From the Equations to the Five CRD Principles

Equations (1) and (2) may help translate some CRD principles [9] into explicit structural assumptions and provide a basis for deriving or testing others. The model incorporates functional complementarity and mutual regulation through differentiated variables and coupling terms. Dynamic balance, transformation, and multiscale recurrence (functional fractality) are stronger claims. They require analyses of trajectories, parameter regimes, responses to perturbations, or explicit cross-scale mappings. The five principles should therefore be treated as connected yet distinct modeling hypotheses.

3.1. Functional Complementarity: Differentiated Functions Relative to a Shared Outcome

The principle of functional complementarity holds that Activator- and Regulator-role processes have distinct yet mutually supportive functions that are jointly relevant to a specified, shared system outcome [9]. In the present model, this differentiation is represented by separate state variables $A\left(t\right)\mathrm{and} R\left(t\right).$ The Activator and Regulator variables are defined relative to a stated system boundary, organizational level, condition, and time scale. An Activator and a Regulator may operate on different time scales, through different mechanisms, and with unequal magnitudes. Their relationship is complementary only if their interaction contributes to the persistence, adaptation, recovery, coordination, or transformation of the defined system. In Equations (1) and (2), this shared relevance is reflected in each variable’s influence on the other’s rate of change and in both variables’ contributions to the system’s trajectory.

Complementarity does not imply harmony, symmetry, equal strength, or simple opposition. Instead, the two roles define and depend on each other. The terms $-bAR,-dAR$ denote such mutual dependence.

In addition, the cross-support terms $+hR\mathrm{and} +kA$ are particularly important for interpreting the CRD as the complementary/transformative elements of the model (see also Section 3.4). At first glance, the term $+hR$ may seem inconsistent with the Regulator’s role. However, regulation need not mean simple suppression. A regulatory process can enable useful activation by restoring resources, reducing harmful excess, coordinating timing, and sharpening the signal-to-noise ratio. It may also help maintain tissue integrity or redirect activity into a viable pathway.

This possibility aligns with established biological control architectures. For instance, appropriate negative feedback can enable an initially strong response while preventing indefinite escalation [13,14]. Resolution pathways can support subsequent immune competence by restoring tissue conditions [11] (see also Section 4). These examples illustrate a general distinction between processes that block activation entirely and those that make activation sustainable, selective, or recoverable [11,15].

The term $+kA$ denotes Activator-driven recruitment or induction of regulation. Depending on the biological context, this relationship may arise when activation triggers negative-feedback mechanisms, inhibitory mediators, and resolution programs [1,2,9]. Other processes that can be recruited include repair mechanisms, compensatory pathways, checkpoint processes, and protective repair responses (see also Section 4).

Thus, the $hR,$ $kA$ terms represent a key CRD hypothesis: activation and regulation can be linked not only by opposition but also by recruitment. This suggests that the seemingly “opposites” may contain each other’s “seeds.” [9] Mathematically, this means the model is complementary/transformative rather than solely antagonistic (see also Section 3.4).

3.2. Mutual Regulation: Reciprocal Coupling Rather than Mere Co-Occurrence

The second principle, mutual regulation, requires demonstrable direct or indirect reciprocal influence between Activator- and Regulator-role processes, not merely the observation that two variables act in opposite directions [9]. These distinctions align CRD with the broader systems-biology emphasis on operationally defined interactions, feedback structures, and context-dependent functions [3,16].

In the present equations, this reciprocal coupling is represented by the regulation and recruitment terms: $-bAR,-dAR,+hR,\mathrm{and}+kA.$ Specifically, the term $-bAR$ models the Regulator-dependent constraint on the Activator variable. This may represent inhibitory feedback, competitive antagonism, adaptive suppression, signal termination, removal of an activating mediator, or another Regulator-dependent restraint.

The term $-dAR$ represents Activator-dependent constraint on the Regulator variable. It recognizes that ongoing activation may alter, deplete, or limit regulatory processes. It denotes a negative contribution to regulatory activity that depends on interaction with the Activator. It can also be a potential reciprocal cost or consumption associated with Activator engagement.

The general principle that activation and regulation involve mutual consumption, costs, or trade-offs is well established [3,17,18]. However, the specific interpretation of $b$and $d$ must remain tied to the proposed mechanism and the measured variables in each application.

The term $+hR$ denotes the reciprocal possibility that regulatory activity supports a productive, recoverable, or appropriately coordinated form of activation. The term $+kA$ adds a second route by which activation influences regulation: Activator activity can recruit a compensatory or inhibitory response [9]. Such recruitment is central to many feedback motifs. Immune activation-resolution interactions are excellent examples [1,2,11]. Thus, these two terms have both mutual support and regulatory meanings.

Mutual regulation in this model is not equivalent to symmetrical mutual inhibition. The coupling may be asymmetric, e.g., $b\ne d,h\ne k.$ One direction may be faster, stronger, delayed, saturated, or absent under certain conditions. Asymmetry is expected in most real systems because activating and regulatory processes often differ in mechanisms, abundance, timescales, and capacities. Therefore, the appropriate test of mutual regulation is not parameter equality but evidence that perturbing one process alters the other’s dynamics in a way consistent with the proposed coupling structure.

3.3. Dynamic Balance: Bounded, Responsive, and Recoverable Trajectories

A system may be dynamically balanced when activation temporarily exceeds regulation, when regulation briefly predominates during recovery, or when variables fluctuate within an adaptive range. Here, dynamic balance denotes a context-appropriate regime in which trajectories remain bounded or recoverable, and the system retains sufficient responsiveness to meet relevant demands [9]. Therefore, the CRD principle of dynamic balance does not mean that $A\left(t\right)=R\left(t\right),$ nor does it require static equilibrium. A healthy system often exhibits structured variability or small oscillations rather than rigid constancy or stillness. Neither excessive flatness nor excessive instability is healthy. Examples include cellular rhythms, hormone secretion, circadian rhythms, homeostasis, allostasis, robustness, and resilience [3,4,5,6,14,15].

The terms related to mutual regulation and complementarity ($-bAR,-dAR,+hR,\mathrm{and}+kA$) may directly contribute to dynamic balance by supporting responsive stability. The terms $-e{A}^3 \mathrm{and} -g{R}^3$ represent nonlinear self-limitation and are also structurally relevant to this principle. These two terms can prevent indefinite growth and represent saturation, resource limits, nonlinear damping, toxicity, depletion, or other amplitude-dependent constraints that prevent unbounded escalation. They can also indicate receptor desensitization or other mechanisms that disproportionately limit high activation or high-regulatory states.

Cubic self-limitation is a tractable phenomenological approximation that can yield bounded trajectories and support the study of nonlinear stability, multiple equilibria, and threshold-dependent changes [7,10]. When a system’s biochemical or physiological mechanism is known, a saturating or mechanistically derived function, justified by the data and biological context, may be preferable.

Whether the system exhibits dynamic balance depends on the full set of parameters, initial conditions, and external inputs. Extensions of the present model that include time delays or additional nonlinearities may also display recurrent or oscillatory dynamics. Feedback loops do not automatically confer stability. Depending on their gain, delay, and nonlinearity, they can instead cause instability or sustained oscillations [14,19].

Therefore, researchers can assess dynamic balance using defined observables. Relevant measures may include response amplitude, peak timing, overshoot, recovery rate, sensitivity to perturbation, and the size of the basin of attraction around a viable state. In translational settings, these properties may be more informative than a single measurement of an Activator’s or Regulator’s magnitude. They distinguish a well-regulated transient response from persistent dysregulation or nonresponsive suppression [9].

3.4. Transformation: Parameter-Dependent Regime Change and Reorganization

The transformation principle concerns qualitative changes in system state, coupling, functional role, or organization. In the present model, transformations can arise when gradual changes in parameters or inputs alter the system’s qualitative behavior. For example, increasing intrinsic Activator gain $a$, reducing the regulatory constraint $b$, weakening Activator self-limitation $e$, or altering Activator-to-Regulator recruitment $k$ may shift the system from rapid recovery to prolonged activation. Conversely, excessive regulatory gain or suppression may shift the system toward reduced responsiveness or persistent inhibition.

In nonlinear systems, such changes may be continuous or abrupt. An equilibrium may lose stability, two equilibria may emerge or disappear, or the system may transition between alternative stable states. Bifurcations can describe such qualitative changes [7,10]. A system may remain in one regime until it crosses a threshold and may not return simply by reversing the original parameter change. Certain biologically important transformation phenomena may require extending the model.

Transformation may also involve changes in functional roles. For instance, the terms $+hR\mathrm{and}+kA$ indicate that, under certain circumstances, such as extremes beyond ordinary thresholds, the roles can change. This may explain the common phenomenon that “when things reach an extreme, they inevitably reverse course.”

For instance, a process initially serving a regulatory role may become an Activator of a new outcome, or an Activator may become regulatory at a higher organizational level. The immune system provides abundant examples [1,2,9,11]. Such reassignment requires a new, explicit definition of the outcome, system boundary, time scale, and causal-relationship modeling. The current model describes the dynamics of a specified A-R assignment but does not eliminate the need for system-specific biological interpretation. Extensions to the current minimal model may also be needed to fully represent these conditions.

3.5. Functional Fractal-like Recurrence: Nested and Cross-Scale AR Architectures

The fifth CRD principle, functional fractal-like multiscale recurrence, holds that related functional A-R architectures may recur across organizational levels. Here, functional fractality does not require exact geometric self-similarity, identical parameter values, or literal mathematical fractals [9]. It more modestly refers to the recurrence of an abstract functional motif in which activating processes are coupled with processes that constrain, coordinate, restore, or redirect.

The two-variable system can be viewed as a single local A-R module: $(A_1,R_1).$ At another level, the same functional architecture may be represented as $(A_2,R_2).$ One module’s output may influence another’s input, parameters, boundary conditions, or outcomes, thereby forming causal bridges [9] between the two levels.

For example, molecular activation and regulatory feedback may shape cellular behavior. Cellular processes may influence tissue-level inflammation and resolution [1,2]. Tissue-level dynamics may contribute to organism-level physiological or clinical states. Systems biology has long emphasized that nested feedback loops and motifs operating across multiple scales can shape network behavior [20,21]. However, cross-scale recurrence in CRD requires explicit mapping via causal bridges to enable a more accurate systemic understanding [9].

The equations in this paper may serve as a prototype for nested modeling [9]. A future multiscale formulation might couple cross-scale A-R modules via parameters, inputs, or network links, for example:

$${\displaystyle \frac{d{A}_i}{dt}}=F_{A_i}\left(A_i,R_i,{\sum }_j{w}_{ij}{A}_j,{\sum }_j{v}_{ij}{R}_j\right) ,$$

(3)

$${\displaystyle \frac{d{R}_i}{dt}}=F_{R_i}\left(A_i,R_i,{\sum }_j{q}_{ij}{A}_j,{\sum }_j{z}_{ij}{R}_j\right),$$

(4)

where $i$ and $j$are module indices, and the coupling parameters $w, v, q, z$represent cross-module influences. Such extensions would allow CRD theory to be compared with established models while keeping the requirement that every proposed A-R relation be empirically specified and independently testable.

3.6. Summary: A Hierarchy of Testable Claims

Table 2 summarizes the equation components and maps them to the five CRD principles. Complementarity is represented by distinct Activator and Regulator roles tied to a shared modeled outcome. Mutual regulation is illustrated by reciprocal restraint and recruitment terms. Dynamic balance reflects bounded, responsive, and recoverable trajectories. Transformation is a derived or extended property that involves threshold-dependent changes in stability, coupling, state, or role assignment. Multiscale recurrence (functional fractality) is a structural hypothesis about nested or cross-scale organizations [9] and requires explicit mapping.

4. Discussion: Potential Applications in Systems Biology and Dynamical Systems Medicine

Systems biology seeks to explain how interacting biological processes generate trajectories over time, including stable states, adaptation, oscillation, and recovery [9,21,22]. Dynamical systems medicine extends this perspective to diseases and treatments [9,22]. It asks not only which molecular components are abnormal but also how a patient’s state changes, why trajectories diverge among individuals, and when intervention is most likely to be effective. This shift adds temporal organization, feedback, and state dependence to traditional interpretations of disease mechanisms or target-based medicine [9,22,23].

The equations here provide a minimal functional description that can be tailored to specific systems. Table 3 presents examples of potential applications of the CRD model in systems biology. The model’s translational premise is that patient trajectories differ not only because components change, but also because coupling, regulatory recruitment, and recovery capacity vary. Timing and susceptibility to transition are also critical.

4.1. From Static Measurements to Patient-Specific Dynamical Biopatterns

Much of clinical medicine still relies on static or intermittently measured variables, such as laboratory values, imaging findings, symptom scores, genomic variants, or disease categories [9]. These measurements are indispensable, but they may not adequately capture the trajectories of a dynamic system. Two patients with similar biomarker values may differ substantially in response amplitude, recovery rate, regulatory reserve, and the probability of transitioning to a more severe state. Identifying dynamical biopatterns may contribute to personalized medicine [9,22].

Disease progression and treatment response reflect interacting molecular, cellular, physiological, behavioral, and environmental processes rather than isolated lesions. In CRD, a dynamical biopattern is defined not only by a variable’s current value but also by its temporal dynamics and its response to perturbations [9]. Relevant observables include amplitude, latency, overshoot, recovery time, hysteresis, switching probability, and resilience.

Within the CRD framework, these measurements can distinguish candidate patient-level states or biopatterns. For instance, these may include excessive or persistent activation; delayed, weak, or depleted regulation; and excessive regulatory suppression or nonresponse (see Table 4) [9]. Other possible dynamical biopatterns include compensated but vulnerable stability, delayed feedback with overshoot or rebound, and transition-prone states near a critical threshold [9].

Thus, patient-specific dynamical biopatterns can be identified by a patient’s Activator-Regulator trajectories rather than by a single biomarker threshold. For example, a persistently elevated inflammatory marker may indicate an ongoing activating drive, failed resolution, persistent exposure, tissue damage, altered clearance, medication effects, or the timing of measurement. The CRD model can guide which possibilities to test and help identify potential dynamical biotargets for disease treatment [9].

4.2. Disease as a Trajectory, Not Only a State

The dynamical perspective is critical when diseases involve recurrent episodes, progressive loss of reserve, abrupt decompensation, incomplete recovery, or heterogeneous treatment response. A patient may appear clinically stable and compensated, yet show slow recovery, greater variability, or heightened sensitivity to additional disturbances. However, assessments that rely solely on average biomarker values or cross-sectional group comparisons may miss these states.

In the CRD model, a system may occupy a stable yet functionally costly state. It may also approach a threshold where a small additional perturbation produces a substantial change in trajectory. These conditions align with earlier work on dynamical diseases [19,23]. Diseases may involve changes in the stability, periodicity, or regulatory organization of physiological systems rather than simply shifting isolated variables outside normal ranges [23]. More recent systems medicine approaches likewise emphasize that diseases can involve network perturbation, loss of resilience, and reorganization across scales [33].

The CRD model may help classify such trajectories based on a defined Activator-Regulator relationship. For example, a transition-prone state might involve insufficient regulatory recruitment after activation, progressive loss of self-limitation, delayed regulatory feedback, or increasingly unstable coupling between the Activator and the Regulator. These are potential hypotheses for comparison and testing.

4.3. Intervention as State-, Timing-, and Sequence-Dependent

A major implication of dynamical systems medicine is that treatment effects may depend on the system’s state at the time of treatment [9,22]. An intervention that benefits early activation may be ineffective or harmful after a transition to chronic persistence, excessive suppression, or regulatory depletion. Likewise, the same intervention can have different effects depending on whether it is applied before, during, or after the recruitment of compensatory processes. These conditions are critical to personalized medicine.

In the CRD equations, an intervention may alter the state variables, their coupling, or their effective parameters. These mathematical changes can help formulate competing mechanistic hypotheses about how an intervention might alter a trajectory. For example, reducing Activator gain may be appropriate in a disease state such as chronic inflammation. In another state, restoring regulatory recruitment may be more effective [9]. The model therefore highlights which parameters change, in which state, when, and with what downstream compensation. Such methods may make personalized medicine more applicable.

Timing and sequence are especially important in systems with delayed feedback [9]. For example, if a treatment suppresses $A$before a necessary response occurs, it may impair adaptation. If it is administered after a transition to a persistent high $A$ state, a more substantial or combined intervention may be needed. If an intervention abruptly increases $R$, it may produce rebound or nonresponse, depending on the values of $h$ and $g$, as well as the system’s state. These possibilities align with the general control-theoretic principle that intervention effects depend on system dynamics, delays, feedback gains, and initial conditions [10]. The CRD model can integrate and enrich these existing models while adding meaningful detail.

4.4. From Molecular Data to Patient-Level State Estimation

Modern systems medicine increasingly combines longitudinal molecular, physiological, imaging, behavioral, and clinical data. The variables $A\left(t\right)$ and $R\left(t\right)$ could be estimated from multiple measurements, for example:

$$\mathbf{y}\left(t\right)=H\left(\begin{array}{c}A\left(t\right)\\ R\left(t\right)\end{array}\right)+\mathit{\epsilon }\left(t\right),$$

(5)

where $\mathbf{y}\left(t\right)$ is a vector of observed changes of clinical variables, $H$ is a measurement map, and $\mathit{\epsilon }\left(t\right)$ represents measurement error and unmodeled variation.

For example, in a defined inflammatory condition, $A\left(t\right)$ might be estimated from a combination of cytokine activities, cell-state markers, tissue injury measures, and symptoms (see examples in Table 3). $R\left(t\right)$ might be estimated from resolution mediators, repair markers, regulatory cell activities, and recovery-related clinical measures. In a metabolic setting, analogous variables might be estimated from longitudinal data on glucose, insulin, hepatic output, activity, diet, and treatment. The validity of this approach depends on whether the measurements truly inform the latent variables and whether the available data can identify the parameters.

4.5. Scope and Limitations: Modeling Assumptions and Interpretation Boundaries

The preliminary model here makes several assumptions that should be clarified. First, it treats time as continuous. This approximation is most appropriate when the relevant processes change smoothly relative to the temporal resolution of observation. Second, $A\left(t\right)$ and $R\left(t\right)$ are treated as aggregate continuous variables. If the system involves few molecules, discrete events, stochastic switching, or sparse cell populations, a stochastic formulation may be more appropriate. Third, we can initially treat the parameters as constant over the analyzed time interval. In biological and clinical systems, however, parameter values may change with development, circadian timing, environmental exposures, treatment, disease progression, or prior system history.

Fourth, Equations (1) and (2) do not include explicit time delays, external inputs, spatial structure, or measurement noise. These omissions are intentional at this stage because the model is designed to identify the consequences of basic coupling and self-limitation before adding further complexity. Later analyses may treat delayed, forced, stochastic, spatial, and network-coupled formulations as principled extensions rather than competing alternatives to the basic model.

Specifically, stochastic and uncommon events may cause abrupt imbalances rather than following regular trajectories. For example, structural injury, toxin exposure, genetic loss-of-function, social conditions, or unmeasured environmental factors may dominate certain conditions. Even when these relationships and dynamics apply, the appropriate model may require multiple variables, spatial structure, stochasticity, or explicit multiscale mechanisms.

CRD equations may help bridge systems biology and dynamical systems medicine. However, clinical translation also faces substantial practical challenges, including patient heterogeneity, comorbidities, prior treatments, evolving environments, and tissue-specific biology. Additional factors include sparse measurements, missing data, uncertain causal direction, and cross-organ compensation. These challenges underscore the need to keep empirical evidence, CRD interpretation, and CRD-specific predictions distinct.

5. Conclusions and Future Directions

In conclusion, this paper has developed preliminary nonlinear equations for the CRD theory, serving as a minimal representation of an Activator–Regulator (A-R) pair:

$${\displaystyle \frac{dA}{dt}}=aA-bAR+hR-e{A}^3,$$

(1)

$${\displaystyle \frac{dR}{dt}}=cR-dAR+kA-g{R}^3.$$

(2)

These equations may formalize a simple yet potentially useful proposition. Within a defined boundary and relative to a specified outcome, Complex Adaptive Systems (CAS) may involve processes that promote, amplify, or propagate, and processes that constrain, restore, or terminate. The model represents intrinsic activation and regulation, reciprocal constraint, Activator-driven recruitment of regulation, Regulator-supported restoration of productive activation, and nonlinear self-limitation.

This formulation is not meant to reduce living systems to two fixed components. $A\left(t\right)$ and $R\left(t\right)$ are outcome-relative functional variables that may summarize different entities, processes, or networks across scales. Their interpretation must therefore specify the focal outcome, system boundary, organizational level, time scale, conditions, and measurable coupling. A molecular mediator, cell population, physiological process, or behavioral response may serve as an Activator in one context and a Regulator in another. Researchers can pre-specify and empirically justify such role changes rather than assigning them retrospectively. Depending on parameter values and extensions, the model may exhibit recovery, persistent activation, excessive suppression, stable imbalance, delayed overshoot, switching, or multistability [7,10].

The equations may help translate the five CRD principles into progressively stronger modeling claims. Complementarity is represented by differentiated, interdependent functional roles that contribute to a shared outcome. Mutual regulation is captured by reciprocal constraints and recruitment terms. Dynamic balance does not require $A=R.$ Rather, it can be demonstrated through bounded, responsive, and recoverable trajectories. Transformation requires evidence that changes in state, coupling, parameters, or external conditions produce qualitative reorganizations, such as switching, hysteresis, bifurcation, or altered functional roles. Multiscale recurrence requires explicit bridge variables and causal evidence across levels, as repeated terminology or visual similarity alone is insufficient.

The terms $+hR\mathrm{and}+kA$ are especially important. They allow the model to capture a more complex relationship than simple mutual inhibition. Regulation may constrain harmful escalation while restoring conditions for useful activation, and activation may recruit regulatory processes that later limit, resolve, or redirect it. Whether these terms improve explanation in a given system is an empirical question. Researchers can justify their inclusion with data and compare them with alternatives.

The appropriate research path is comparative and falsifiable. Candidate A-R systems should be defined in advance. Longitudinal and perturbational studies may help provide evidence for coupling and assess parameter identifiability and uncertainty. The model’s potential contribution to systems biology is therefore practical. It can help organize candidate variables, formulate hypotheses, and identify useful longitudinal observables. It may also help compare alternative explanations for conditions such as persistence, recovery, overshoot, and transition.

CRD’s potential contribution to dynamical systems medicine is similarly pragmatic and integrative. If researchers can measure or estimate Activator and Regulator variables over time, the model may help distinguish patient trajectories, such as excessive activation or maladaptive suppression (see Table 4). These distinctions could inform hypotheses about intervention timing, treatment sequence, rebound risks, and recovery capacity. These proposed applications may help integrate and enrich mechanistic systems biology, network medicine [33], quantitative systems pharmacology [34], causal inference, and clinical testing. Thus, CRD may help make personalized therapy and dynamical systems medicine more achievable.