Index analysis: An approach to understand signal transduction with application to the EGFR signalling pathway

In systems biology and pharmacology, large-scale kinetic models are used to study the dynamic response of a system to a specific input or stimulus. While in many applications, a deeper understanding of the input-response behaviour is highly desirable, it is often hindered by the large number of molecular species and the complexity of the interactions. An approach that identifies key molecular species for a given input-response relationship and characterises dynamic properties of states is therefore highly desirable. We introduce the concept of index analysis; it is based on different time- and state-dependent quantities (indices) to identify important dynamic characteristics of molecular species. All indices are defined for a specific pair of input and response variables as well as for a specific magnitude of the input. In application to a large-scale kinetic model of the EGFR signalling cascade, we identified different phases of signal transduction, the peculiar role of Phosphatase3 during signal activation and Ras recycling during signal onset. In addition, we discuss the challenges and pitfalls of interpreting the relevance of molecular species based on knock-out simulation studies, and provide an alternative view on conflicting results on the importance of parallel EGFR downstream pathways. Beyond the applications in model interpretation, index analysis is envisioned to be a valuable tool in model reduction.


Introduction
Large scale models of biochemical reaction networks are increasingly used to study the dynamical response of a system to a specific input.The response is often measured in terms of some output quantity of interest.Examples include the therapeutic inhibition of the epidermal growth factor underlying characteristics of the system.In time scale separation, the system dynamics needs to present different time scale, where a partitioning into slow and fast states and a subsequent quasi-steady state approximation of the fast variables results in the reduced model [12].In balanced truncation [13], the basic idea is to transform the system into its principal components and neglect the least important components while still maintaining the same input-response behaviour as the original system.Balanced truncation has seen limited application for signalling networks [5].The authors conclude that there is no "one size fits all" model reduction technique as each approach exploits different model characteristics, consequently there have been various attempt to combine different techniques [5].Profile likelihood based model reduction utilises parameter identifiability to designate likely parameter candidates for reduction [10].It is a powerful tool for model reduction, focussing on experimental data and parameters, rather than on states variables and time.There appear, however, some interesting links between their four scenarios and our proposed indices.In [14], we introduced the concept of an empirical input-response index by building and expanding on concepts from control theory (empirical controllability and observability gramians).Given some input and response, the empirical index quantifies the relevance of a state variable for the given input-response relationship by some finite set of perturbations.We illustrated in a clinically relevant setting, how the empirical index can guide us to reduce a large systems biology model of the blood coagulation network by eliminating states variables, either by completely removing them from the system or by assuming them to be constant.
In the present article, we substantially extend the concept of an index and at the same time focus on understanding a given signal transduction network rather than reducing it.This index analysis is first introduced based on a number of simple model systems to ease understanding of the different index types and their application.We finally illustrate the power of index analysis in application to a large-scale model of the EGFR signalling cascade [1,15].This signalling network has been intensively studied in the context of tumour genesis and tumour progression as well as for anti-cancer treatment strategies, e.g., to overcome drug resistance in non-small cell lung cancer [16].From a signal transduction point of view, the EGFR network remains a challenging system due to the large number of interacting molecular species and its multiple February 20, 2023 4/41 parallel pathways.

Methods
Many dynamical models in systems biology/pharmacology are of the form dx dt (t) = f x(t); p , x(t 0 ) = x 0 + u 0 (1) with x(t) ∈ R n denoting the vector of state variables at time t ∈ [t 0 , T ], and p ∈ R m denoting the vector of parameters.The function f : R n × R m → R n represents the reaction kinetic model, and is typically of the form with stoichiometric vectors ν µ ∈ R n and reaction rates α µ ∈ R of the reactions µ = 1, . . ., M .
The function h : R n → R q maps x(t) to the output y(t) of interest, often only a single state variable.The initial condition x(t 0 ) comprises two parts: (i) the state x 0 ∈ R n of the system prior to the stimulus, and (ii) an input or stimulus u 0 ∈ R n at time t 0 .The solution of the ordinary differential equations (ODEs) in eq. ( 1) is written as with state transition function Φ t,t 0 .The output then takes the form The assumptions on f (time-independence) and h (independent on parameters p) can be relaxed.
In the sequel, we used both, z k and [z] k to denote the kth entry of a vector z = (z 1 , . . ., z n ); the February 20, 2023 5/41 choice depended on whatever was deemed easier to read.An analogous notation was used for matrices.
Input-response index.
according to eq. (1).To quantify the extent to which a perturbation of the input impacts a given state variable, we considered a perturbed input u per = u ref + ∆u per and quantified the impact using a first-order Taylor approximation (indicated by the dot on top of the "=" sign) with Jacobian Relative to the reference input, this resulted in a perturbation Thus, we next quantified the extent to which a perturbation of the kth state variable at t * impacts the output y on the remaining time interval.The key idea is to reinterpret the perturbation ∆x k (t * ) as an input to the model system in eq. ( 1) with (unperturbed) initial condition This resulted in the perturbed output (to first order Taylor approximation): we thus quantified the resulting perturbation on the jth output component as The definition is motivated from control theory (see also below).The first factor represents the time-average integrated impact of an (infinitesimal) state perturbation over the time span rather than the integral should be considered.For time-dependent inputs, also the second factor becomes an integral; for delta-type inputs as in our case it reduced to a single value at the initial time.
In general, ir k (t * ) is a matrix of dimension (number of outputs)×(number of inputs).For the common situation of a single state input (i.e., u 0 has only a single, say ith non-zero entry) and a single response state (i.e., y(t) = h(x(t)) = x r (t) ∈ R for some state index r = i), the input-response index is real-valued and can be written in terms of two local sensitivity coefficients with sensitivity coefficients It is a distinct feature of the input-response index that it combines both, the impact of the input on a state variable as well as the impact of the state variable on the output.Neither one nor the other on its own is in general informative to quantify the relevance of a state variable for a February 20, 2023 8/41 given input-response relationship.In a control theoretical setting, the factors can be interpreted as a controllability index C k and an observability index O k .
To ease comparison of the ir-indices for different points in time, we normalised each index by the sum of all indices, resulting in the normalised ir-indices (nir): As a result, the nir-index takes only values between 0 and 1.The sum of ir-indices is also of interest, as it gives some indication on the overall magnitude of the ir-values.Details on the computation and the extension to time-dependent input or fixed output time of the ir-indices can be found in the Supplementary Material S2&3 Text.
We classified a state variable as dynamically important, if its input-response index is above a (user-defined) threshold at some point in time However, rather than perturbing the kth state variable at t * and simulating with the original system of ODEs, now the state variable is unperturbed and the system of ODEs is modified (details below).In both cases, the difference to the reference solution is quantified on the remaining time interval [t * , T ].Up to time t * , the modified system coincides with the reference system, so no additional factor appears in the definition in eq. ( 15) (in contrast to the second factor in eq.10).We finally note that the normalised indices can take values larger than 1.
We defined four state classification indices; they aim at classifying a state from the perspective of their effect on the output.
• The environment index (env k ) quantifies to what extent the kth state variable can be classified as an environmental state, defined as being constant in time.Thus, the modification of the system of ODEs at time t * is to set the right hand side (RHS) of the kth ODE to zero.Thus, for the partially neglected index pneg k , the modification to the system of ODEs is to set all reaction rate constants to zero that involve the kth state variable as reactant species.In many reaction kinetic systems, this can easily be realised by setting x k (t) = 0 for all t ∈ [t * , T ].For the completely neglected index cneg k , the modification to the system of ODEs is to set all reaction rate constants to zero that involve the kth state variable as reactant or product species.
The state classification indices only measure the impact (of a specific change of the system of ODEs) on the output.For a final classification of a state variable, it is, however, important to also measure the impact on the state variable itself.Thus, for the state classification indices env k and qss k , we additionally defined a corresponding relative state approximation error In all examples, we successfully used a threshold of 0.1, consistent with the threshold of 10% for the ir-index.Fig. 1 summaries the strategy we used to classify states.

Material
We chose a number of simple model systems to illustrate the application, interpretation and usefulness of the indices, of which one is included in the Results section and the remaining are included in the Supplementary Material S4 Text.
We then studied the epidermal growth factor (EGF) receptor signalling network [1,15,17] to illustrate how the indices allow to obtain detailed insights into the dynamic behaviour of complex, large-scale systems biology/pharmacology models.The EGFR system is an an important pathway in cell division, death, motility and adhesion [1,18,19].In addition, it is of key interest in the development of anti-cancer therapies, as the pathway is often dysfunctional in tumour cells.We used a detailed model of the EGFR reaction network [15] consisting of 106 state variables and 148 reactions.We followed [15] regarding the abbreviations of state variable names.
The original model includes a lumped pseudo state of degradation products that serves as a substitute for various individual degradation products.To allow for a more refined analysis of receptor degradation, we modified the original model by separating the degraded receptor species (EGF-EGFR * ) 2 -deg into six separate degradation products, increasing the number of state variables by six to 112.This extension does not change the remaining system dynamics.
All initial conditions and parameter values for the model were taken from [15, Suppl.Table 2].
In addition, all corrections reported in [20] were taken into account.
The model was implemented in Matlab 2021b and will be uploaded to the permanent and open-access repository zenodo.In contrast to our expectations, the system published in [15] is not in steady state in the absence of EGF, i.e., the stimulus of the system.Some state variables do change, including EGFR, EGFRi, Grb2, Sos and Grb2-Sos; see Supplementary Material S2

Index analysis for illustrative model system
To illustrate the index analysis in application, we first considered a simple reaction cycle model sketched in Figure 2. In the model, the signal A transforms B into C, which in turn activates D.
In addition, A, C and D may be subject to degradation.In this example, A is considered the input and D the response variable.We distinguished three different scenarios defined by the parameter values in Table ??.These were chosen to illustrate how the indices allow to analyse and differentiate between the scenarios.We use Scenario 1 to introduce the indices in detail and build on this in Scenarios 2 and 3. We used the decision-tree in Figure 1 to guide the analyses.Figure 3A shows the time course of the state variables, while Figure 3B shows the normalised ir-indices.Here and in all other analyses, we empirically (as with many thresholds) used a threshold of 10%; this choice was supported a-posteriori by the results.One nicely observes from panel B that the dynamic importance of state variables changes substantially over time: Initially, the signal A has the largest dynamic importance; it decays, however, very quickly and below a threshold of 10%.This threshold has been chosen empirically (as for most thresholds).
In   threshold for all times.The bottom-right panel illustrates the impact of modifying the reaction system to enforce C being in partial steady state from time t * = 0 onwards.As can be seen, the solution of the modified system (dashed lines) is a very good approximation to the solution of the reference model (solid lines).For sake of illustration, the bottom left panel show the impact of modifying the system so that B is environmental (i.e.constant) from t * = 0 onwards.Clearly and in line with the large environmental index of B, the resulting approximation is very poor.
Thus, we conclude for Scenario 1 that C can be considered in partial steady state, while all other states are considered as dynamically important.
For Scenarios 2+3, Figure 5  For Scenario 2, we conclude that all states are dynamically important.For Scenario

Input-response indices of the EGFR signalling pathway guide subsequent analysis of the key molecular species
To illustrate applicability and usefulness to large-scale systems, we next performed an index analysis of the epidermal growth factor receptor (EGFR) signalling cascade.The EGFR pathway is activated by binding of EGF to EGFR.Dimerised EGF-EGFR autophosphorylates and recruits a series of adaptor molecules called GAP, Grb2 and Sos.Signal transduction may occur via membrane-bound or internalised species; in addition it occurs via two major pathways: the Shc-dependent and the Shc-independent pathway.The two pathways, however, do not act February 20, 2023 20/41 independently from each other, since there are many molecules involved in both pathways.Both pathways eventually activate Ras-GDP, a well-known oncogene and the merging point of both pathways.Subsequent activation of Raf transduce the signal to the MAP kinase cascade and finally to ERK.The output signal is double-phosphorylated ERK, which transiently increases as a response to the input stimulus.In view of the input (EGF) and the output (ERK-PP), the signalling cascade is sometimes termed the EGF-ERK-PP system.A graphical representation can be found in the Supplementary Material S1 Fig.
While the principal transduction of the signal is well known, the relative importance of the different pathways (Shc-dependent and Shc-independent, membrane-bound vs. internalised forms) and its molecular constituents is still not well understood.To index-analyse the EGFR signalling cascade, we chose a constant extra-cellular EGF stimulus of 50 nM and a time interval [0, 100] min, as in [15].Figure 7A shows the temporal response of the ERK-PP output signal; the peak of the signal is reached 3 min.The sensitivity-based indices were determined according to eq. (11).Figure 7B shows the sum of the ir-indices over February 20, 2023 21/41 time.We clearly identify three different phases: an initial peak (0-0.3min), a short high plateau (until 3min); followed by an extended and finally decaying low plateau.While the extended low plateau corresponds to the decay phase of the output, the initial peak and the short high plateau correspond to signal onset and steep increase Figure 8 shows all normalised ir-indices that exceed a threshold of 10% at least once during the time span-20 out of a total of 112.The threshold value was chosen empirically, but also matched a gap in the decay of the maxima of the normalised input-response indices (see Figure 8D).In broad terms, Figure 8 shows an ordered appearance and disappearance of states, as might be expected.A closer examination reveals that at any point in time there exist often 3-5 states with a normalised ir-index above 10%, but always at least one index (see Supplementary The state just below the threshold is (EGF-EGFR * ) 2 -GAP-Shc * -Grb2-Sos.
Further observations can be gained from Figure 8: First, not all species that are absolutely necessary to transmit EGF to ERK-PP have a normalised ir-index exceeding 10%, including adaptor proteins Shc, Grb2, Sos as well as Raf and MEK.Second, while Phosphatase3 is known to be involved in signal deactivation, its nir-index already peaks around 1.4 min and thus during February 20, 2023 23/41 signal activation.Phosphatase 1 and 2, however, do not seem to have a similar role during the activation phase.Finally, MEK-P and ERK-P-MEK-PP seem to be relevant during the deactivation phase (see Figure 8C), while we would rather associate them with the activation phase.The analyses below are guided by these observations.

Phosphatases1-3 show very different behaviours and functions
Phosphatase3 (abbreviated P'ase3, when part of a complex) is involved in de-phosphorylation of ERK-P and ERK-PP (see also Figure 9).Figure 10A depicts the time course of key molecular species involved in the local Phosphatase3 network during signal activation.Once the kinase MEK-PP is present, it phosphorylates ERK via ERK-P to ERK-PP.Rather than observing an increase in ERK-P followed by ERK-PP, however, we first see a steep increase of ERK-P:P'ase3, February 20, 2023 24/41 indicating that Phosphatase3 immediately binds ERK-P to form a complex and subsequently de-phosphorylates it.Only when the level of Phosphatase3 decreases sufficiently, ERK-P levels increase markedly.As with ERK-P, double-phosphorylated ERK-PP is immediately bound by Phosphatase3 and subsequently de-phosphorylated, as can be inferred from the step increase of ERK-PP:P'ase3.Again, only when the Phosphatase3 level further decreases, the output signal ERK-PP increases to high levels.Thus, Phosphatase3 delays signal onset by sequestering ERK-P and ERK-PP; moreover, it controls the ERK-PP peak concentration as well as signal deactivation.This is confirmed by simulating the model with no Phosphatase3, as shown in Figure 10B.In summary, the described action of Phosphatase3 can be understood as a protection mechanism against activation of ERK-PP by spurious or random phosphorylation of ERK or ERK-P in the absence of a signal.
For Phosphatase1, Figure 11A show the normalised state classification indices.Since both, the env-index and the corresponding relative state error (not shown) are below the threshold, we classify Phosphatase1 as environmental.A further look at the Phosphatase1 levels of the reference model (see Figure 11B) is confirmatory.In addition, the prediction of the modified model with constant Phosphatase1 levels from t = 0 on (i.e., as environmental state) are shown; the differences to the original model predictions are very minor.
We coloured all state variables classified as environmental in pink in the model scheme in

Raf * dynamics is fast throughout activation and deactivation
The maximal value of the ir-index of Raf * is 0.17% 10% (see Supplementary Material S3 We coloured all state variables classified as partial-steady state in light green in the model scheme in Figure 9.We infer that several other species, including a large fraction of internalised species, act on an instantaneous time scale.

Ras-GTP * recycling is important for signal prolongation
The cycle of Ras-GDP activation and Ras-GTP inactivation is a central motif of the signalling cascade (see Figure 9).While Ras-GDP and -GTP have nir-indices above the threshold, the inactivated form Ras-GTP * has a nir-indices below the threshold for all times.Figure 12A shows the state classification indices for Ras-GTP * .Since none of them is below the threshold for all times, no further classification is possible (see also Figure 1).

Ras-GTP levels
Prot-mediated internalization and subsequent degradation of internalised receptor species plays an important role in output signal shut-down.Figure 9B shows six internalised EGFR species including degradation (out of 16 degradation reactions in total).For these six receptor species, degradation has a large impact on the output signal.Figure 13 shows the prediction of the reference model and a modified model with no degradation of these six species from t = 0 (realised by classifying the corresponding degradation products, e.g., (EGF-EGFR * ) 2 -deg as completely negligible).Without degradation of these receptor species, the signal is strongly prolonged: An elevated concentration of internalised receptors increases the recycling rate to the membrane, impacting the total concentration of membrane-bound receptor species.While the cytosolic species Ras-GDP is hardly changed, elevated receptors levels eventually increase the activation rate from Ras-GDP to Ras-GTP via increased levels of (EGF-EGFR * ) 2 -GAP-Shc * -Grb2-Sos.Increased levels of activated Ras-GTP, finally, result in a prolonged output signal.
February 20, 2023 30/41 bound and internalised pathway is again the "dynamic nature of the state variables".No internalised species is classified as dynamic (large nir-index), while a large number is classified as partial-steady state.Using the partially/completely neglected index, we identified the relevance of receptor degradation from specific internalised receptor species (indicated by degradation reactions in Figure 9).Absence of receptor degradation from these species increases the pool of internalised receptors and thus receptor recycling to the membrane; this eventually results in higher levels of activated Ras-GTP and finally in signal prolongation.At the same time, we identified the role of internalised MEK in sequestering Phosphatase2 in complexes, lowering the deactivation rate of Raf * and thereby prolonging the signal.
A detailed study of the EGFR signalling pathway is presented in [15].The authors quantified the relevance of reactions based on the concept of impact control coefficients, and the importance of proteins based on a fractional change of their total concentration.The time-dependent output was described by three characteristics: the amplitude, duration and integral of the ERK-PP profile.Index analysis allows a complementary view on the system.It focusses on individual state variables and on time, in contrast to reactions and lumped total concentrations.In [26] general principles that govern signal transduction are identified, with the central conclusion that collectively, kinases control amplitudes more than duration, whereas Phosphatases tend to control both.Our time-resolved index analysis of the Phosphatases of the EGRF signalling network, in particular Phosphatase3, supports this conclusion and adds an additional detail: Phosphatases might also control the time to signal onset.In addition, we identified mechanistic principles underlying the conclusion including, e.g., the stoichiometric ratio of the Phosphatases and their binding partners (see paragraph on Phosphatases1-3 in the Results).While free Phosphatase1 levels are hardly impacted by complex formation with Raf * , free Phosphatase3 levels are reduced by three orders of magnitude; nearly 100 % of Phosphatase3 is sequestered in complex with ERK-PP for a long time.
Overall, the input signal EGF exerts its impact on the output ERK-PP only during a very small time window (see Supplementary Material S14 Index analysis naturally links to model reduction.In [14], we introduced an empirical inputresponse index by building and expanding on the concepts of controllability and observability from control theory.Based on the empirical index, we proposed an iterative model reduction scheme.In application to the blood coagulation network, we illustrated its usefulness in a clinically relevant setting.A key feature of the proposed model reduction technique is its reference to a local regime in the state space (by defining a reference input in addition to input/output state variables).We demonstrated the advantage of such a local approach by identifying different reduced models based on different reference inputs.For the given application, this allowed to understand the lack of impact of certain genetic modifications for the outcome of the standard blood coagulation test and the presence of impact for a modified test.In the present work, the focus is rather on understanding a given signal transduction network.At the same time, the index analysis provides new insights and opens new possibilities for future model reduction approaches.By introducing state classification indices jointly with corresponding relative state approximation errors, we clearly discriminate between the impact of an approximation on the output and on the state itself.The newly introduced state classification indices provide further options for the model reduction approach proposed in [14].
Index analysis may contribute to the identification of promising new drug targets in the future by providing a deeper understanding of the pharmacologically targeted system.Moreover, we envision that index analysis will be beneficial to study the difference between a healthy and diseased state, e.g., by comparing the index analyses of the two conditions and identifying state variables that change their input-response index or state classification index.
February 20, 2023 35/41 All in all, we believe that the proposed index analysis approach substantially broadens our means to analyse and understand complex signal transduction models in systems biology.

•
The partially and completely neglected indices (pneg k , cneg) quantify to what extent the kth state variable can be neglected and thus removed from the system.This can be done in two different ways: Consider the reaction A − − B − C. When neglecting C, we might either want to partially remove C from the system and maintain its 'producing' reaction (thus maintaining the degrading reaction of B), resulting in A − − B − * , or we might want to completely remove C and all reactions involving it, resulting in A − − B.

Figure 1 .
Decision tree for state classification based on indices.See text for details.

Fig.
Fig. To ensure comparison to the original publication in[15], however, we did not make any further changes.

Figure 2 .
Figure 2. Simple reaction cycle model and scenario-specific parameter values.(A)Reaction network, and (B) parameter values for the three scenarios, for which the model system was studied.The time span was t ∈ [0, 0.1] min, the input A, and the response variable D. The initial conditions were identical in all three scenarios: (A 0 , B 0 , C 0 , D 0 ) = (2, 100, 5, 0) nM.Units: k on in 1/nM/min; all other reaction rate constants in 1/min.

Figure 3 .
Figure 3. Scenario 1 of the simple reaction cycle model: time course of state variables and normalised indices.Time course of state variables (left) and normalized ir-indices (right) for the model specified in Figure 2.

Figure 4 .
Figure 4. Scenario 1 of the simple reaction cycle model: Relative state approximation error and modified dynamics.A: State classification indices for state B and B: for state.C: Relative state approximation errors for B and D: for C. E: Comparison of reference dynamics and modified dynamics from t * = 0 (dashed lines) for B as environmental state and F: for C in partial steady state.For details on the model, see Figure 2.
shows the time course of all state variables and the normalised input-response indices.For Scenario 2 (left column), we conclude from Figure 5C that all states are classified as dynamically important, since no nir-index is below the threshold for all times.In contrast, for Scenario 3, we conclude from Figure 5D that only states A and D are dynamically important, while states B and C are not.Figure 6 (top) shows the state classification indices for B (left) and C (right).Two indices are below the threshold for all times: the env-index for B and the pss-index for C. The bottom panel finally confirms this classification.The relative state approximation errors for B as environmental state (left, solid blue line) and C as in partial steady state (right, dashed red line) are below the threshold for all times.

Figure 5 .FebruaryFigure 6 .
Figure 5. Scenario 2 & 3 of the simple reaction model: A & B: Time course of state variables and C & D: normalised ir-indices for Scenario 2 (left) and Scenario 3 (right).For details on the model, see Figure 2.

Figure 7 .
Figure 7. Output signal and sum of ir-indices evolving over time A: Transient increase of ERK-PP (output) in response to the EGF (input) stimulus.Inset zoom: signal activation occurs within 3 min.B: Sum of ir-indices over time, showing three phases: initial peak (0-0.3 min), high plateau (0.3-3 min), and low plateau including decay (3-100 min)

FebruaryFigure 8 .
Figure 8. Normalised ir-indices over time and ordered maximal value.A, B and C: Normalised ir-indices for the three phases (0-0.3 min, 0.3-3 min, 3-100 min).Shown are all 20 indices with a maximal value exceeding 10%.D: Order of decay of maximum value of the normalised ir-indices.All 20 states above the threshold of 10% are membrane-bound or free cytosolic forms and (as far as applicable) belong to the Shc-dependent pathway-as opposed to internalised forms and the Shc-independent pathway; see Supplementary Material S3 Table.The state just below the threshold is (EGF-EGFR * ) 2 -GAP-Shc * -Grb2-Sos.

Figure 9 .
Figure 9. Schematic with state classification of the signal transduction network focussing on the Shc dependent pathway, including the 20 state variables with largest maximum input-response index (light blue, see Figure 8D), environmental state variables (purple), state variables in quasi-steady state (green) and further state variables (dark blue).States being part of the membrane-bound and internalised pathway are coloured orange in panel (B).The red boxes mark the input and output state variables; coloured dots as part of reaction arrows indicate intermediate complexes.A similar graphic for the Shc-independent pathway is given Supplementary Material S13 Fig.

Figure 9 .FebruaryFigure 10 .Figure 11 .FebruaryFigure 12 .
Figure 9.We infer that only one other species-Prot, a coated pit protein that mediates receptor internalisation-is classified as environmental.

FebruaryFigure 13 .
Figure13.Analyses of receptor degradation.Comparison of the output ERK-PP and Raf * for the reference simulation (solid lines) and a modified model (dashed lines) with no degradation from six internalised receptor species (those with degradation reaction in Figure9, realised by classifying the degradation products as cneg).
Fig, left); it does so by strongly impacting, i.e., February 20, 2023 34/41 controlling, state variables during this time frame.In combination with roughly seven orders of magnitude smaller observability indices (see Supplementary Material S15 Fig, right), this can be interpreted as some robustness property of the signalling cascade.Random fluctuations of constituents in the absence of an input signal are unlikely to span several order of magnitude needed to spontaneously activate the cascade.
[t * , T ] on the output; the integral form guarantees that a transient impact during the time span is taken into account, even if it occurs only during some short time span in the interval [t * , T ].If, however, only the output at a single event time t event is of interest, then [J y (t event , t * )] j,k While B and D reach interim values of 45% around 2 min, C increases only up to 10%, before it starts to decays again.It stays below 10% for the entire time span.The dynamic importance of B and D continue to evolve with almost opposite behaviour.Eventually, B decays and D increases to its maximal value.For a classification of states, the particular features of the index are not considered relevant; the key feature considered is whether an index stays below the threshold for all times, or not.For the input-response index, we considered a state dynamically unimportant, if its ir-index stays below the threshold of 10% for the entire time span.Otherwise, we consider it as 'not unimportant', i.e., as dynamically important.Thus, we considered A, B and D as dynamically important, but not C.
contrast, the indices of all other state variables are negligibly small initially and increase February 20, 2023 14/41 steeply.

Table ) ,
a surprisingly small number given the relevance of Raf * in signal transduction.Figure 12A & B show the normalised state classification indices and the corresponding relative state .Since both, the pss-index and the corresponding relative state error are below the threshold, we classified Raf * as partial-steady state.The C panel shows the prediction of the reference model and a modified model with Raf * in partial-steady state from t = 0; the profiles are indistinguishable.Thus, signal propagation through Raf * is 'instantaneous'. *