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Organization at criticality enables processing of time‐varying signals by receptor networks

2020; Springer Nature; Volume: 16; Issue: 2 Linguagem: Inglês

10.15252/msb.20198870

1744-4292

Angel Stanoev, Akhilesh P. Nandan, Aneta Koseska,

Neurobiology and Insect Physiology Research

Article24 February 2020Open Access Transparent process Organization at criticality enables processing of time-varying signals by receptor networks Angel Stanoev Angel Stanoev Department of Systemic Cell Biology, Max Planck Institute for Molecular Physiology, Dortmund, Germany Search for more papers by this author Akhilesh P Nandan Akhilesh P Nandan Department of Systemic Cell Biology, Max Planck Institute for Molecular Physiology, Dortmund, Germany Search for more papers by this author Aneta Koseska Corresponding Author Aneta Koseska [email protected] orcid.org/0000-0003-4263-2340 Department of Systemic Cell Biology, Max Planck Institute for Molecular Physiology, Dortmund, Germany Search for more papers by this author Angel Stanoev Angel Stanoev Department of Systemic Cell Biology, Max Planck Institute for Molecular Physiology, Dortmund, Germany Search for more papers by this author Akhilesh P Nandan Akhilesh P Nandan Department of Systemic Cell Biology, Max Planck Institute for Molecular Physiology, Dortmund, Germany Search for more papers by this author Aneta Koseska Corresponding Author Aneta Koseska [email protected] orcid.org/0000-0003-4263-2340 Department of Systemic Cell Biology, Max Planck Institute for Molecular Physiology, Dortmund, Germany Search for more papers by this author Author Information Angel Stanoev1, Akhilesh P Nandan1 and Aneta Koseska *,1 1Department of Systemic Cell Biology, Max Planck Institute for Molecular Physiology, Dortmund, Germany *Corresponding author. Tel: +49 2311332225; E-mail: [email protected] Molecular Systems Biology (2020)16:e8870https://doi.org/10.15252/msb.20198870 PDFDownload PDF of article text and main figures. Peer ReviewDownload a summary of the editorial decision process including editorial decision letters, reviewer comments and author responses to feedback. ToolsAdd to favoritesDownload CitationsTrack CitationsPermissions ShareFacebookTwitterLinked InMendeleyWechatReddit Figures & Info Abstract How cells utilize surface receptors for chemoreception is a recurrent question spanning between physics and biology over the past few decades. However, the dynamical mechanism for processing time-varying signals is still unclear. Using dynamical systems formalism to describe criticality in non-equilibrium systems, we propose generic principle for temporal information processing through phase space trajectories using dynamic transient memory. In contrast to short-term memory, dynamic memory generated via “ghost” attractor enables signal integration depending on stimulus history and thereby uniquely promotes integrating and interpreting complex temporal growth factor signals. We argue that this is a generic feature of receptor networks, the first layer of the cell that senses the changing environment. Using the experimentally established epidermal growth factor sensing system, we propose how recycling could provide self-organized maintenance of the critical receptor concentration at the plasma membrane through a simple, fluctuation-sensing mechanism. Processing of non-stationary signals, a feature previously attributed only to neural networks, thus uniquely emerges for receptor networks organized at criticality. Synopsis Cells continuously integrate time-varying signals using surface receptors. We propose a dynamical mechanism for signal integration through critical system's organization, resulting in a dynamic transient memory, generated via “ghost” attractor – a remnant of previous activity. Receptor networks use memory to process time-varying growth factor signals. Memory must be dynamic. “Ghost” of a saddle-node provides a dynamical mechanism for memory in receptor activity. Recycling could provide self-organized maintenance of the critical organization through a fluctuation-sensing mechanism. Introduction In a wide variety of biological processes including embryogenesis, immune cells motility, wound healing or cancer metastasis, cells sense and interpret time-varying chemical signals that reflect the non-stationary environment to which they readily adapt. It has been, for example, demonstrated that time-varying growth factor signals not only trigger corresponding phenotypic output in cells, but a range of input frequencies can bias towards a specific function (i.e. differentiation), irrespective of growth factor identity (Ryu et al, 2015). Cells can also direct their motility through continuously changing patterns of chemical signals such as travelling waves of chemoattractants (Skoge et al, 2014), using memory of stimulus history to integrate conflicting signals (Foxman et al, 1999; Welf et al, 2012). Generally, a transient memory of stimulus history is a main requirement for systems that process time-varying signals, as a means to integrate temporal dependencies inherent in the signal (Hochreiter & Schmidhuber, 1997; Maass et al, 2002). How cells sense the growth factors from their environment has been extensively studied using equilibrium and non-equilibrium descriptions of sensing through ligand binding/unbinding dynamics for stationary levels of receptors and ligands (Berg & Purcell, 1977; Bialek & Setayeshgar, 2005; Wang et al, 2007; Rappel & Levine, 2008; Endres & Wingreen, 2009; Mora & Wingreen, 2010). These studies provide analysis of the fundamental limits of ligand concentration sensing by direct mapping to receptor occupancy that serves as a proxy for receptor activity. However, these mapping properties cannot satisfy and thereby do not apply to systems where memory requirements are necessary for integrating time-varying signals. Receptor activity dynamics, on the other hand, is not only influenced by ligand binding dynamics, but rather reflects the dynamics of the biochemical network in which the receptor is embedded (Stanoev et al, 2018). Non-trivial dynamical solutions can hereby emerge, in particular due to recurrent interactions between the network components (Reynolds et al, 2003; Tischer & Bastiaens, 2003). In a broad range of biological systems, positive feedback interactions give rise to bistable dynamics, which is considered to underlie memory features (Xiong & Ferrell, 2003; Wang et al, 2009; Burrill & Silver, 2010; Doncic et al, 2015). However, signal-induced switching between basal and high receptor activity states, and thereby permanent memory formation, limits response to upcoming stimuli (Stanoev et al, 2018). To overcome equivalent limitations of stable states, information processing in the context of real-time computations of sensory stimuli by neural microcircuits, universal frameworks using transient dynamics and state-dependent trajectories have been proposed (Maass et al, 2002; Durstewitz, 2003; Jaeger & Haas, 2004). These formalisms typically contain high-dimensional state representations and non-linear intrinsic activation dynamics of the neuron components (Maass et al, 2002; Jaeger & Haas, 2004; Ozturk & Principe, 2005) and thus cannot be directly translated to biochemical networks. Therefore, a conceptual framework that describes processing of time-varying signals on the level of cellular sensing networks is lacking. We propose here a saddle-node (SN) “ghost” as a minimal dynamical mechanism that enables processing of time-varying growth factor signals. Critical organization in a vicinity of a SN bifurcation enables transient memory of receptor activity to be realized via the metastable “ghost” state. In contrast to the short- or long-term memory that stem from stable attractors, we demonstrate that this transient memory is dynamic and thereby uniquely promotes integrating and interpreting complex temporal growth factor signals. A clear distinction between a transient memory that reflects a dynamical state to a kinetic relaxation of receptor activity in terms of the signal integration capabilities is also shown. Using single-molecule reaction–diffusion simulations on the other hand, we depict how such dynamic memory can be realized on molecular level. Based on the experimental findings that the epidermal growth factor receptor (EGFR) system operates at critical organization (Stanoev et al, 2018), we propose a fluctuation-sensing mechanism as a basis for self-organized maintenance at the critical region and discuss its limitations. We further discuss why organization at criticality represents a generic dynamical mechanism which enables processing of time-varying growth factor signals by cell surface receptors. Results Organization at criticality enables transient temporal memory of growth factor signals to be manifested in receptor activity Systems that sense time-varying signals require memory in order to integrate the signal information (Hochreiter & Schmidhuber, 1997). A minimal cellular sensing network that accounts for memory in receptor activity (Ra) is a two-component toggle switch (Fig 1A), where the double-negative feedback (DNF) interaction (Reynolds et al, 2003; Tischer & Bastiaens, 2003; Stanoev et al, 2018) between the active receptor and an inactivating enzyme (e.g. a phosphatase), PDNF,a, follows the law of mass action: (1) Figure 1. Memory manifestation depending on parameter organization Diagram of a two-component toggle switch between active receptors (Ra) and the deactivating enzyme, protein PDNF,a. Input—fraction of ligand-bound receptors (LRa). Molecular details described in Materials and Methods. Bifurcation diagram of the R-PDNF toggle switch, depicting Ra response with respect to PDNF,T/RT, in the absence of input. Shading: blue—monostable region, magenta—vicinity of the saddle-node (SN) bifurcation point and green—bistable region. Solid/dashed lines—stable/unstable steady states. Two-parameter (LRa, PDNF,T/RT) bifurcation diagram depicting the parameter space where bistability exists (green area). Vertical lines denote organization in irreversible bistable (green), critical (magenta), reversible bistable (yellow) and monostable (blue) organization.*—cusp bifurcation. Steady-state receptor activity response for increasing input doses in the different organizations. Solid/dotted lines—stable/unstable steady states. Arrows—switch on/off points. Temporal receptor activity responses to step-wise modulation of the input (LRa, grey) for organization in the distinct parameter regimes. Data information: In (D, E), the colours correspond to the respective system organization depicted with vertical lines in (C). Download figure Download PowerPoint The system combines autocatalytic receptor activation and mutual inhibition mechanisms (Fig 1A) that govern the protein state transitions between the active (Ra, PDNF,a) and inactive (Ri, PDNF,i) states of the switch components. They are described in further detail in Materials and Methods with the corresponding parameters. Bistability in receptor activity is exhibited between two saddle-node bifurcation points for a broad range of the bifurcation parameter—the PDNF,T/RT concentration ratio—in the absence of stimulus input (Fig 1B, green shaded region). The two stable states correspond to the basal and the high receptor activity states. The system maintains bistability also for a certain range of inputs (Fig 1C, green shaded region). The effective input for the cells in this case is the fraction of ligand-bound receptors (LRa) that reflects the extracellular ligand concentration (Materials and Methods). Since the temporal receptor activity dynamics upon changes in growth factor stimulation is governed by the receptor dose–response dynamics, it will also depend on the system's organization in parameter space (the PDNF,T/RT concentration ratio). We therefore investigate next whether the stable attractor solutions could underlie emergence of transient memory in receptor activity. For PDNF,T/RT concentration ratio that corresponds to organization in the bistability region in the absence of growth factors (Fig 1C, green line), the numerical simulations show irreversible receptor activation. This is reflected both, in the receptor's steady-state response to changes of growth factor doses (Fig 1D, green) and in the temporal receptor activity profile (Fig 1E, green) upon step-wise modulation of growth factor input (Fig 1E, grey). The system thereby exhibits a temporal long-term memory to the presence of single growth factor stimulus: the receptor activity is irreversibly maintained at high levels after growth factor removal. In contrast, for system's organization in the reversible bistable regime (Fig 1C, yellow line), the receptor's activity response displays hysteresis with respect to the input doses that activate/deactivate it (Fig 1D, yellow). This induces a short-term memory only regarding the growth factor dose that activates the system, but the memory is not reflected in the temporal receptor activity profile (Fig 1E, yellow). Thus, the receptor activity is not prolonged in time upon removal of the growth factor input. Temporal response without prolonged receptor activity after input removal was also observed for PDNF,T/RT concentration ratios that correspond to organization in the monostable regime (Fig 1C and E, blue lines). In this regime however, there is no hysteresis and thus no memory of the growth factor dose that activates the system (Fig 1D, blue). These results indicate that both the long- and the short-term memory that result from the presence of stable attractors do not fulfil the conditions necessary for processing time-varying inputs. The long-term memory is not transient and thereby it will inhibit responsiveness to upcoming cues (Stanoev et al, 2018), whereas the short-term memory only corresponds to the growth factor dose that activates the system and is not reflected in the temporal receptor activity profile. For organization in the vicinity of the saddle-node bifurcation point (Fig 1B and C, magenta), the numerical simulations demonstrate the presence of memory of the dose that activates the receptor (Fig 1D, magenta). The activation of the receptor, similarly as in the reversible and irreversible bistable regimes, occurs in a switch-like manner at a threshold input dose, indicating that spurious activation is filtered out. However, additionally, high receptor activity was transiently maintained over time after removal of the growth factor (Fig 1E, magenta). This shows that critical organization confers to the sensing system a transient memory of the previous input-driven activation. Saddle-node “ghost” as a dynamical mechanism of transient temporal memory To understand how transient memory occurs for critical organization of the system in the vicinity of the saddle-node bifurcation point, and in particular how it is distinguished from short- and long-term memory, we studied qualitatively the dynamical Ra-PDNF,a behaviour. We analysed how the phase space trajectories evolve in relation to the changes in the geometry of the underlying phase space as a function of a pulsed stimulus. Generally, the relative positioning of the nullclines, which are determined by the system parameters, shapes the phase space geometry. In non-autonomous or input-driven systems, either the geometry of the underlying phase space can be altered (change in the positioning, shape and size of the attractors), or its topology (change in the number or stability of the attractors) (Verd et al, 2014; Jimenez et al, 2017). We also estimated the associated quasi-potential landscapes (Strogatz, 2018; Fig 2; Materials and Methods; and Verd et al, 2014) that depict the energy-like levels associated with the states. The phase space trajectories flow downhill the landscapes, towards the valleys defined by the stable steady states. Figure 2. Qualitative Ra-PDNF,a behaviour with respect to phase space changes to pulsed input Left: receptor response (blue) to single growth factor pulse (yellow) and respective profile of ligand-bound receptors (LRa, grey) for positioning in the monostable regime (PDNF,T/RT = 4.3). Inset: responsiveness to subsequent input pulses. Middle/right: phase space diagram and nullclines intersecting at the basal/high activity receptor steady state denoted with blue squares/circles, respectively. Blue arrows: phase space transitions upon administering and removal of stimulus (, , respective time points denoted in left plot). Insets: calculated quasi-potential landscapes. Same as in (A), only for positioning in the bistable regime (PDNF,T/RT = 2.5). Green arrows: phase space transitions (, ). Same as in (A), for positioning at the critical transition between monostability and bistability (PDNF,T/RT = 2.957). Magenta arrows: signal administration () and removal (). The transition and the associated phase space plot demonstrate the existence of a “ghost” attractor. Parameters for (A–C) as in Fig 1. Download figure Download PowerPoint We first consider organization in the monostable regime (Fig 1B and C, blue), where the system does not exhibit any memory in receptor activity (Fig 1E). In this case, the pulsed stimulus induced changes in the phase space geometry of the system (adding/removing stimulus: , blue transitions, Fig 2A), thereby triggering continuous and reversible re-positioning of the single steady-state attractor that captivates the state trajectory. This leads to receptor response that closely follows the input (Fig 2A left and inset). However, when in the absence of stimulus, the system is poised in the valley of basal receptor activity in the double-well quasi-potential landscape characteristic for the bistable organization (Fig 1B and C, green), a topological phase space change where this state vanishes occurs at a threshold signal concentration. This results in a transition to the high receptor activity state (Fig 2B, green transition), which also explains the previously demonstrated switch-like response to increased growth factor doses (Fig 1D, green). Upon signal removal, the reverse topological change leads to re-establishing of bistability ( green transition). However, the trajectory remains in the occupied high activity stable steady state (green circle in Fig 2B middle). Thus, the first pulse will activate the receptors and this will hinder further responsiveness to upcoming stimuli due to the long-term memory that results from this stable attractor organization (Fig 2B left, inset). In contrast, for organization in the reversible bistable regime, the changes in the topology of the phase space induced by the pulsed stimulus allow for reversible switching between the two stable attractors, the basal and high receptor activity (topological transitions are omitted from Fig 2 for clarity). These topological changes thereby also guide the phase space trajectory such that the time spent in the stable attractors is equivalent to the administration time of the stimuli, resulting in the absence of prolonged receptor activity upon growth factor removal (Fig 1E, yellow). When the system is positioned in the vicinity of the saddle-node bifurcation point (Fig 1B and C, magenta), a supra-threshold input pulse induces transition from the basal monostable to the high activity monostable state (Fig 2C, magenta transition) via the bistable region in a switch-like manner (Fig 1D). Upon input removal, these consecutive topological transitions are reversed. However, there is a delay between establishing the single stable attractor (magenta state iii) and the system trajectory converging to it (magenta state iv), resulting in prolonged receptor activity before relaxation to basal level (Fig 2C left). This delay results in a transient memory of receptor activity that does not hinder further responsiveness of the system (Fig 2C inset). The transient memory is a consequence of the critical dynamical behaviour near the SN bifurcation point. In this organization, the nullclines intersect only once, indicating a single stable steady state of basal receptor activity. However, they are positioned very close to one another in the phase space area where the steady state of high receptor activity is stable for bistable organization (compare Fig 2B and C, middle), resulting in a quasi-potential landscape with a very shallow slope (Fig 2C middle, inset). Thus, when the system transits back from the bistable to the monostable region, the remnant of the saddle node that disappeared in this transition generates a metastable state that continues to capture the incoming trajectories (see Movie EV1 for a stochastic simulation of this process). Such delayed dynamics, different from a slow relaxation kinetics, is referred to as a feature of a “ghost” attractor (Strogatz & Westervelt, 1989) and has been previously shown in some driven dynamical systems such as ferroelectrics or semiconductor lasers (Rogister et al, 2003). We have demonstrated here, however, that saddle-node “ghost” serves as a unique dynamical mechanism of transient temporal memory, enabling receptor activity to be maintained at high levels for a limited period of time after growth factor removal. Dynamic temporal memory is a prerequisite for processing time-varying growth factor signals To integrate and interpret the information contained in time-varying growth factor profiles however, the transient temporal memory in receptor activity must be dynamic. In other words, the total duration of the receptor activity should depend on the previous stimulus history. We therefore probed the features of the transient memory resulting from the saddle-node “ghost” using a train of two subsequent growth factor pulses with inter-pulse interval shorter than the duration of the memory. The numerical simulations demonstrated that the receptor activity dynamics could rapidly adapt to the second growth factor pulse, and the period in which the receptor activity is maintained high is longer than in the case of a single pulse stimulation (compare Fig 3A, magenta to Fig 2C, left). In contrast, the time-frame in which the receptor activity was maintained high for the other temporal memory manifestation, organization in the irreversible bistable regime, was equivalent as for a single pulse (Fig 3A, green). The receptor activity profile in the absence of temporal memory on the other hand, such as for organization in the reversible bistable and monostable regimes (Fig 3A yellow and blue, respectively), closely followed that of the growth factor stimuli. Figure 3. Dynamic transient memory uniquely enables processing time-varying growth factor signals Receptor responsiveness to two subsequent 5-minute growth factor pulses for different organizations of the system, denoted by colours as in Fig 1C. Yellow shaded area: growth factor pulse duration. Grey temporal profile: input (LRa). Distribution of total duration of receptor activity (as a fraction of total time, 480 min) calculated for growth factor pulse trains constructed from 12 subsequent 5-min pulses randomly distributed over time. Different colours denote responses for different system's organization (equivalent to Fig 1C). The distributions are generated from 1,000 independent realizations. Distribution of number of disjoint intervals of receptor activity (top black lines in D), estimated for the growth factor pulse trains in (B). Exemplary temporal receptor activity profiles for different growth factor pulse trains realizations for critical organization. Black line segments: disjoint intervals of receptor activity. Yellow shaded area: growth factor pulse duration. Grey temporal profile: input (LRa). Dynamic range of receptor activation for input that activates the system (LRa = 0.15), as a function of . Parameters for (A–E) as in Fig 1. Download figure Download PowerPoint We next simulated 1,000 different non-periodic growth factor pulse trains by randomly distributing twelve 5-min growth factor pulses over period of 480 min. The total duration of high receptor activity, reflecting the degree of history-dependent signal integration, was estimated for the individual realizations. The resulting distributions strongly depended on the organization of the system in parameter space. In the reversible bistable and monostable regimes, the distributions were narrow and closely reflected the total duration of growth factor pulses (Fig 3B, yellow and blue), whereas for irreversible bistable positioning, the receptor activity was constant over all realizations and equivalent to the integration time (Fig 3B, green). In contrast, for positioning in the critical vicinity of the SN bifurcation, the total duration of receptor activity was highly variable (Fig 3B, magenta), reflecting the varying degrees of history-dependent signal integration that depend on the temporal signature of the signal profile. This variability is also depicted by the exemplary temporal receptor activity profiles following different periodic growth factor pulse trains (Fig 3D). For the different growth factor pulse trains, we also estimated the number of disjoint intervals of receptor activity over the integration time (black line segments in Fig 3D, middle), thereby reflecting whether the response follows the temporal partitioning of the signal. Again, in the presence of dynamic memory, the respective distribution was broad (Fig 3C, magenta), emphasizing the varying degrees of partitioning in response to complex signals. In the irreversible bistable organization, a single disjoint interval was observed for all pulse train realizations (Fig 3C, green), since the receptor is irreversibly activated upon the first pulse. In the reversible bistable and monostable organization on the other hand, narrow skewed distributions were identified (realizations between 7–8 and 12 were identified, Fig 3C, yellow and blue). This reflects small degree of partitioning that results from immediate signal re-occurrences within the relaxation time of the system. To increase the degree of integration of a monostable system in general, its kinetic parameters can be tuned to match the relaxation time to the transient memory length (Fig EV1, Materials and Methods). However in this case, a slow decay rate will greatly decrease the number of disjoint intervals in comparison with the one for critical organization, due to the absence of rapid reversibility in the response. Increasing the decay rate on the other hand will diminish the history-dependent signal integration, and the total duration of receptor activity will resemble the total duration of growth factor pulses. Thus, to avoid the trade-off between history-dependent signal integration and adaptation to the temporal partitioning of the signal, the response must exhibit transiently maintained steady levels of high receptor activity, followed by rapid reversibility to basal levels. Hence, simple relaxation kinetics of a monostable system cannot account for processing time-varying signals by receptor networks. Click here to expand this figure. Figure EV1. Information processing features for an exponentially decaying relaxation process Temporal receptor activity profiles for distinct decay rates β (dashed red lines; model and respective parameters in Materials and Methods) as compared to the temporal profile for organization at criticality (magenta, equivalent to Fig 2C). Yellow shaded region: growth factor pulse duration. Corresponding distributions of total duration of receptor activity (as a fraction of total time, 480 min) calculated for growth factor train pulses constructed as in Fig 3B. Distributions of number of disjoint intervals of receptor activity for the different β values. The distributions are estimated from 1,000 different realizations of growth factor pulse trains as in Fig 3. Download figure Download PowerPoint At the critical organization, the emergence of the dynamic memory is additionally complemented with a maximal increase in receptor activity to minimal growth factor dose that activates the system. Scanning the PDNF,T/RT ratio demonstrated that the dynamic range of the response amplitude rapidly increases when transiting from the monostable towards the bistable regime, with a clear peak at the SN bifurcation (right to left, Fig 3E). These results therefore demonstrate that critical organization is crucial for robust responsiveness to and processing of time-varying growth factor signals. Molecular realization of transient memory in receptor activity To understand how transient memory can be realized on a molecular level, we studied how transient receptor activity can be generated and maintained in the absence of stimulus using single-molecule reaction–diffusion simulation framework on a two-dimensional surface (Materials and Methods). By analogy to the main model (Equation 1), single receptor molecules (R) are susceptible to activation, and once active they can propagate their state via direct contact to other inactive molecules. The receptor molecules can become susceptible again following deactivation by the active PDNF molecules, with which they interact in double-negative feedback manner. To characterize the activity dynamics of the system, we use the basic reproduction number R0 (Dietz, 1993) that reflects the transmission potential—the average number of newly activated molecules by a single active receptor molecule in the course of its active lifetime, i.e. before its deactivation. If R0 < 1, the overall receptor activity in the system decays to a basal level (Fig 4A, top), whereas for R0 > 1, the system exhibits supercritical behaviour and the receptor activity propagates in a branching fashion across the cell surface (Fig 4A, bottom). Figure 4. Molecular realization of transient memory Schematic representation of activity propagation in relation to microscopic single-molecule activation/inactivation dynamics. Top: diminished activity for R0 < 1. Bottom: propagated activity for R0 > 1. Basic reproduction number for varyi