Non-canonical Activation of Akt in Serum-Stimulated Fibroblasts, Revealed by Comparative Modeling of Pathway Dynamics
[['Akt , also known as protein kinase B ( PKB ) , promotes cell survival , inhibits apoptosis and contributes to cancer progression , metastasis , and drug resistance [1] .', 'Constitutive activation of Akt is sufficient to cause cancer in mammals [2] .', 'Even subtle or transient effects that increase Akt activation are important to elucidate , because any existing pathways that can heighten Akt activation in normal cells may also be utilized to maintain an excessive activation of Akt in cancer .', 'Many factors can influence Akt pathway function , but the key steps of Aktp308 phosphorylation can be summarized simply .', 'Growth signals and stress pathways activate PI3K ( Phosphatidylinositol 3-kinases ) , which converts PIP2 ( Phosphatidylinositol ( 4 , 5 ) -bisphosphate ) into PIP3 ( Phosphatidylinositol ( 3 , 4 , 5 ) -trisphosphate ) .', 'PIP3 is a membrane phospholipid with potent signaling effects , and its concentration is kept low by the PTEN ( Phosphatase and tensin homolog ) phosphatase , which converts PIP3 back into PIP2 [3] .', 'PIP3 recruits PDK1 ( 3-phosphoinositide-dependent protein kinase 1 or PDPK1 ) to the cell membrane [4] .', 'PIP3 also recruits Akt to the cell membrane , where Akt can be phosphorylated at Thr308 by PDK1 [5] .', 'Once phosphorylated , Akt dissociates from the membrane and targets many downstream effector substrates [6] .', 'Inactivation of Akt by dephosphorylation is regulated by a variety of phosphatatases , with the PP2A phosphatase known to be particularly important for dephosphorylation of Thr308 [7] .', 'We refer to these key steps as the canonical Akt pathway .', 'The total net activation of Akt results from multiple competitions between processes of activation and inactivation .', 'Activation of Akt kinase activity involves phosphorylation of two residues: Thr308 in the activation loop and Ser473 in the C-terminal hydrophobic motif .', 'The phosphorylation of Akt at Thr308 is a prerequisite for full Akt activity , and in some contexts is a better correlate with activity than phosphorylation at Ser473[8] .', 'For example , in tumor samples , the phosphorylation of Thr308 is a more reliable biomarker for Akt kinase activity than Ser473 [9] .', 'Therefore , we focus on Thr308 phosphorylation in this work .', 'When quiescent cells are stimulated with growth factors , the level of Akt phosphorylation rises from very low to a brief extreme of very high phosphorylation , followed by a decline to a moderate steady level .', 'Indeed many signal transduction pathways have a similar trend ( sometimes called “overshoot” or “negative feedback” ) , which is routinely modeled as an effect of receptor internalization [10 , 11] .', 'Although many pathways do not display overshoot [12] , this peak and decline is a documented feature of Akt dynamics [13 , 14] .', 'Mathematical modeling with ordinary differential equations ( ODEs ) is best known for revealing how a set of simple reactions can constitute a complex system with emergent behavior [15 , 16] .', 'Another use of dynamic models is to identify discrepancies between the accepted mechanisms of a pathway , and the experimental observations of pathway dynamics .', 'For example , we used a dynamic model of pathway activation to uncover unknown effects of a drug regulating apoptosis sensitization [17 , 18] , and to uncover a novel negative feedback loop in TGF-β/Smad signaling [19] .', 'Yet another use of modeling is to study an observed behavior with unknown mechanism [20] .', 'In such cases , computational modeling can illustrate multiple alternative hypotheses , as demonstrated by Hua et al . for alternative mechanisms of Bcl-2 action [21] , or by Xu et al . for alternative topologies of Erk activation [22] .', 'The peak and decline of Akt phosphorylation is a rich source of indirect information that computational models may be able to exploit .', 'Previous models of Akt activation dynamics have given limited treatment to the peak and decline behavior .', 'Hatakeyama et al . modeled Akt and Erk activation by assuming that receptor internalization would cause Aktp308 to decline after its initial peak .', 'Some recent studies [11 , 23 , 24] have modeled Akt activation as a part of larger models to study the crosstalk between Akt and other pathways ( e . g . Ras/Raf ) , and did not investigate the overshoot of Akt .', 'Furthermore , few computational models have incorporated new sources of experimental data about Akt dynamics [25] .', 'We focus on Akt overshoot dynamics for several reasons .', 'Firstly , pathway dynamics can give insight into underlying pathway mechanisms .', 'Secondly , the dynamics of Akt phosphorylation may affect its downstream functional effects .', 'Thirdly , the landmark role of Akt makes it particularly important to understand quantitatively .', 'Countless groups measure phospho-Akt , relying on it to reflect the canonical pathway , including PI3K , PTEN , PIP3 , and PDK1 [26] .', 'In this work , we used modeling and experiments to study the peak and decline of Aktp308 following the re-introduction of serum in the culture medium of serum-starved fibroblasts .', 'In the first unit of results , we modeled the canonical Akt pathway downstream of PIP3 and upstream of Aktp308 , comparing model simulations against experimental observations .', 'The canonical model could not explain how the observed PIP3 dynamics would lead to the observed downstream dynamics of Aktp308 .', 'This major discrepancy led us to reject the canonical model .', 'The second unit of results describes a novel strategy for studying the unknown effect; we generated a series of alternative hypotheses by assuming that a non-canonical effect occurs , with arbitrary dynamics , at any one step of the canonical pathway .', 'For each alternative hypothesis , we found an optimized curve shape for the non-canonical effect , by determining how much deviation would be necessary at each timepoint , in order for the model to fit the data .', 'In the third unit , computational simulations suggested an experiment to differentiate among the alternative hypotheses , and two hypotheses with gross violations of the new experiments were ruled out .', 'In the fourth unit , we predicted , performed , and interpreted another experiment and further narrowed the range of plausible alternatives .', 'This iteration between wetlab and drylab allowed us to conclude that the peak and decline of Akt phosphorylation at Thr308 is predominantly caused by a non-canonical effect recruiting or retaining Akt at the cell membrane .'], ['There is published evidence for many other mechanisms to influence Akt .', 'Akt activation can be affected by Tcl1 [32] , ILK [33] , PIKE [34] , and PAK1 [35]; by chemical modifications such as ubiquitylation [36] , phosphorylation by CK2 [37] , phosphorylation by Ack1 [38] , disulfide formation [39] and O-GlcNAcylation [40]; and by regulatory influences such as lipid rafts [41] , reactive oxygen species [42] , SHPS-1 [43] , TTC3 [44] , CTMP [45] , and ATP-binding [46] .', 'In addition , Akt binds over 100 known target substrates [47] , any of which could exert a localization or scaffolding effect .', 'Another list could be made for regulators of PDK1 or PP2A .', 'From the vast array of non-canonical mechanisms that could affect Akt phosphorylation , we do not know which are applicable to our context; many of these effects have been identified in cancer cells with mutated Akt behavior , or in specialized cell types , or under extreme conditions .', 'Rather than designing an alternative hypothesis for each published biochemical effect , we instead categorized the universe of possible non-canonical effects according to which step of the canonical pathway is affected .', 'That is , we established one alternative hypothesis for each step of the canonical pathway .', 'We refer to this approach as a top-down strategy for coverage of alternative hypotheses .', 'Categorizing hypotheses by their “point of intervention” will allow us to accept or reject hypotheses depending on the dynamics of the effect .', 'Starting with six steps of the pathway corresponding to six alternative hypotheses , we immediately discarded the hypothesis of perturbed PDK1 recruitment to the membrane , based on our time-series measurements of membrane PDK1 [29] , and repeated in S1 Fig . PDK1 recruitment dynamics were a close mirror of the PIP3 dynamics , indicating no non-canonical effect perturbing or delaying the PDK1 dynamics .', 'For the remaining 5 alternative hypotheses S2 Fig ) , we do not know the molecular mechanism that would cause each hypothetical deviation , so we cannot model the alternatives with explicit enzyme kinetics .', 'Instead , we constructed an arbitrary spline curve to represent the magnitude of the hypothetical deviation over time .', 'The shape of the spline curve can later be optimized for the model to fit the observed trends ( see “Using splines to represent non-canonical deviations” and subsequent sections in the Methods section ) .', 'In other words , optimizing the perturbed model to fit the data would solve for the shape of the spline curve , which tells us when the deviation would occur .', 'Each spline curve was integrated into the canonical model by adding a pseudo-reaction ( Fig 3 ) at the appropriate step of the pathway .', 'The pseudo-reaction should not be misinterpreted as representing a true biochemical mechanism; it is only an interface between the spline curve and the differential equations of the network model .', 'Using splines and pseudo-reactions as described , we converted the five alternative hypothesis into five computational models ( Fig 3 ) : Akt recruitment could be altered through a PI3K/PIP3-dependent mechanism ( “PIP3-dependent recruitment model” , M1 , Fig 3A ) , Akt recruitment could be altered by a mechanism that is independent of PI3K/PIP3 ( “PIP3-independent recruitment model” , M2 , Fig 3B ) , Akt translocation from membrane back to cytosol could be blocked or delayed ( “retention model” , M3 , Fig 3C ) , or Akt dephosphorylation could be disrupted ( “dephosphorylation model” , M4 , Fig 3D ) , or the phosphorylation of membrane-localized Akt could be altered ( “phosphorylation model” , M5 , Fig 3E ) .', 'Fig 3F shows a flowchart for how each model , with its non-canonical effect , was fit against the data ( see also the “parameter estimation” sections of Methods ) .', 'Our approach to managing the alternative hypotheses was first to study how the hypotheses could be differentiated from each other .', 'This led to a small number of follow-up experiments .', 'After each experiment , any hypothesis that grossly violated the observed trends was ruled out .'], ['Aktp308 phosphorylation clearly exhibits overshoot behavior , but the time-scale of the overshoot was considerably later for Aktp308 than for PIP3 ( Fig 1 ) .', 'In the first unit of our results , the canonical model was unable to explain the delay between the peaks of PIP3 and pAkt ( Fig 2 ) , suggesting that some non-canonical effect causes the overshoot dynamics of Aktp308 .', 'A critical consideration for the interpretation of Fig 2 is parameter estimation , which is never guaranteed to find globally optimal values .', 'If our estimated parameters are significantly worse than optimal , then a correct hypothesis could appear to violate the observations .', 'In other words , our search is incomplete .', 'Incompleteness of search is an obstacle to claiming that a model can’t fit a dataset .', 'This is relevant when ruling out the canonical model , and also when claiming that altenative models can’t satisfy the data .', 'To address this risk , we pursued multiple complementary methods of parameter estimation and a large number of random restarts before accepting a high-violation model to be “best-fit” to the data .', 'Secondly , we avoided ruling out models unless they showed gross violations of the observed trends .', 'Third , we could sometimes find an underlying reason or explanatory intuition for why the failure would be unavoidable .', 'Future work could improve our confidence , with less need for human judgment , when testing whether any parameters exist for a given model to match a given behavior .', 'After finding the canonical model to be inadequate , we adopted a novel top-down strategy for exploring alternative hypotheses: we assumed there was a single non-canonical effect responsible for the majority of the deviation , and we systematically enumerated the steps of the canonical pathway as the possible places where the deviation could occur ( Fig 3 ) .', 'Furthermore we developed a novel spline-based method for explicitly representing the time-course of an unknown non-canonical deviation .', 'We inserted non-canonical deviations , one for each step of the pathway , to create a set of alternative models .', 'Each non-canonical devation is encoded using a pseudo-reaction driven by a pseudo-species with unknown time-course .', 'The model ( with unknown time-course for the pseudo-species ) was optimized for best match with the data .', 'In essence this approach carries out the following assumption: the observed Akt curve ( s ) should be equal to the network-propagated effects of PIP3 , plus the network-propagated effects of the non-canonical deviation .', 'NetworkPropagate ( PIP3→ ) +NetworkPropagate ( deviatio→n ) =pAkt→ .', 'Because we knew the pAkt curve and the PIP3 curve , we attempted to solve for the curve of the unknown deviation , which means solving for the values of the spline knots .', 'Values for the spline knots were obtained using stochastic global optimization methods for parameter estimation of the model .', 'Parameter estimation adjusted the values of the spline knots and all other variable parameters , many times in iteration ( Fig 3F ) , to maximize agreement with the observations .', 'The use of an explicit curve to encapsulate the unknown deviation provided a complete model of the perturbed system , capable of being simulated ( Box', '1 ) and scored ( S1 Tables ) .', 'Simulations could then illustrate possible implications and side-effects of each alternative hypothesis .', 'When a model does fit experimental data , we should not over-interpret the model as describing how the alternative hypothesis would necessarily work .', 'It is merely an illustration of how it might work .', 'This distinction arises because the parameter estimation process may encounter multiple minima , while having no fundamental basis for preferring one minimum over another .', 'The global minimum is not necessarily the ground truth ( even if the ground truth were a single entity , and even if we could identify the global minimum ) .', 'Fitting single models allows us to conclude simply that there exists at least one parameterization allowing this hypothesis to fit the data .', 'Fitting a family of models with incrementally different parameterizations is potentially more powerful , but in preliminary trials it yielded results similar to single-model fitting ( S1 Document ) .', 'For future work , when more datapoints are available so the variance can be estimated , we hope to rank alternative models by quality of fit to the entire experimental distribution , similar to the work of Vyshemirsky et al . with Bayesian ranking of models [49] or the work of Eydgahi et al . with Bayesian scoring of parameterizations [50] .', 'To distinguish among alternative hypotheses , two sets of follow-up experiments were pursued .', 'In unit 3 of the results , membrane fraction time-series excluded the phosphorylation model and the dephosphorylation model because they couldn’t explain why Akt would remain localized at the membrane after PIP3 levels peak and decline .', 'In unit 4 , LY29 time-series experiments showed that the non-canonical effect is dependent on PI3K ( Fig 7 ) , leaving little possibility for PIP3-independent recruitment to be correct .', 'Finally we applied LY29 to constitutively activated cells and found that the non-canonical presence of Akt at the membrane remained detectable even when PIP3 was no longer sufficient to recruit PDK1 to the membrane ( Fig 8 ) .', 'Although recruitment and retention may co-exist in some equilibrium , this experiment shows that retention is more accurate than recruitment to characterize the effect .', 'Table 3 summarizes which models were considered viable before and after each stage .', 'Many common experimental methods measure only relative fold-change and not absolute abundance .', 'Furthermore , measurements contain many sources of error , which do not necessarily follow normal distributions .', 'For example , immuno-blots generally deviate farther from an ideal linear response when the protein has very low abundance or very high abundance .', 'The standard deviation of the mean is a useful measure of error , but is potentially misleading in time-series experiments because there is greater consistency in the timepoint-to-timepoint trends for each batch , than there is from batch to batch in replication of the time-series experiment .', 'There are statistical approaches for analysis of grouped measurements , but such approaches are problematic when combined with other sources of error .', 'Instead we reduced the observed datasets to a low-resolution representation of curve shape , using three types of information: the time of peak abundance , the width of the peak , and the fold-change from peak to steady-state .', 'For plotting , we used the median rather than the mean ( Fig 5 ) and to communicate the spread of the data , we have published primary data from all replicates .', 'Each step is rigorous but we refer to the process as semi-qualitative because the curve-shape features have lower resolution ( fewer parameters ) than the original dataset , and because the decision which alternative hypotheses to reject included human inspection of outliers , rather than pure automation .', 'However , the steps could be performed using greater automation .', 'The curve-shape scoring is automated , and deciding which models are in gross violation of the curve shape scoring could be performed by thresholding .', 'Several aspects of this study may interest Akt specialists .', 'One contribution of this work is analyzing the temporal offset between PIP3 dynamics and Akt dynamics , although the existence of this offset is not novel [4 , 6 , 26 , 51] .', 'Our more important contribution for Akt biology is finding that Akt is retained and the membrane after its upstream activators decline .', 'The following are some known non-canonical effects that might mediate Akt retention at the membrane: ( 1 ) Akt can bind PI34P ( phosphatidylinositol ( 3 , 4 ) -bisphosphate ) in addition to PIP3 [52] , and so PI34P might be responsible for the prolonged localization of Akt at the membrane after PIP3 levels decline .', 'If PI34P levels are dependent on dephosphorylation of PIP3 , then PI34P levels would also be blocked by LY29 , satisfying the criteria of PI3K-dependence .', '( 2 ) Pak1 can serve as an anchor for localizing Akt at the plasma membrane [35] .', 'The Pak1 effect would likely be PI3K-dependent because PI3K activates Rac1 , which activates Pak1 .', '( 3 ) NHE1 is an integral membrane protein that promotes Akt signaling through an indirect scaffolding effect [53] .', 'The antiporter activity of NHE1 is known to be PI3K-dependent [54] , suggesting that its Akt scaffolding function might also be PI3K-dependent .', '( 4 ) Membrane microdomains and caveolae are known to regulate Akt in a localized manner [55] , and are associated with many aspects of lipid signaling including PI3K .', 'Experimental testing and characterization of individual non-canonical factors is beyond the scope of this work , but will be the subject of future experiments .', 'Despite our assumptions , more than one significant non-canonical effect may be present .', 'For example , we excluded the hypothesis of altered dephosphorylation ( M4 ) because it wasn’t sufficient by itself to explain the data , but some type of altered dephosphorylation [48] could very well occur in conjunction with another mechanism .', 'Another possibility we didn’t model is that a single non-canonical phenomenon could affect more than one step of the model .', 'For example , pH could affect many steps of the pathway [56] .', 'Finally , there could be other violations of our simplifying assumptions: for example , we interpreted the lack of PP2A in the membrane fraction to indicate that dephosphorylation is cytosolic in this cell type .', 'Another reason why we prohibited models with multiple non-canonical deviations is because each hypothetical effect ( linear spline ) increases the number of unknown parameters in the system , and worsens the risk of overfitting .', 'When a model has too many unknown parameters , relative to the amount of data available for identifying the parameters , the model can become so under-determined that almost any hypothesis can fit the data , and nothing can be ruled out .', 'On the other extreme , if we require the parameters to be identifiable from the data [57] , then our model will become overly simplistic .', 'Our modeling presents a case of under-determined models with 26 unknown parameters .', 'Interestingly , the models were still useful , despite being under-determined ( and implicitly , overfit ) , for a very important reason: overfitting can only lead to errors with incorrect acceptance of an untrue model , not erroneous rejection of a true hypothesis .', 'Despite potential generosity to false hypotheses through overfitting , we were still able to reject most of the hypotheses after iteration with experiments .', 'The dynamic curve shape of Akt activity is important because mechanisms allowing normal cells to achieve a transient peak of extreme Akt activation could be mutated to provide a permanent state of extreme Akt activation in cancer .', 'Akt is one of the most consistently over-activated pathways in cancer [58] .', 'Another reason why the timing of Akt activation is important is because “hub” proteins ( i . e . , factors with many downstream targets and effectors ) often achieve differential functions through the dynamics of activation , and not exclusively through the magnitude of activation [59 , 60] .', 'This occurs for NF-kappaB [61] , MAPK [62 , 63] , p53 [64] , and TGF-β/Smad signaling [65 , 66] .', 'The effects of Akt on cell proliferation and cell cycle progression [67] have been found to depend on the timing , not just the magnitude , of Akt activation .', 'We do not know the downstream functional implications of Akt overshoot dynamics in these cells , but it is logical to presume that membrane versus cytosolic localization of activated Akt could affect specificity .', 'Akt has many targets at both membrane and cytosol , and altering the localization of activated Akt would alter the availability of active Akt toward different subsets of targets .', 'Textbook-level simplifications of signal transduction are necessarily incomplete , but to improve a model beyond the textbook level confronts an information overload , not only from the massive published literature , but also from the associated unknowns ( e . g . , reaction rates ) that accompany each published discovery [68] .', 'Exploratory experiments , chosen ad hoc , would consume enormous resources .', 'The goal of our work was to use computational modelling within the most fundamental layer of biological decision-making: namely , constructing hypotheses and deciding what to test .', 'Our chief result was to simulate alternative hypotheses ( and their non-canonical deviations ) , and to use these simulations for investigating a temporal offset between upstream and downstream species ( PIP3 and Aktp308 ) in Akt activation .', 'Computational modeling clarified what information would be gained by each experiment , and extracted stronger inferences from the analysis of existing data .', 'In the future , when hypothesis-evaluation becomes more automated , top-down coverage of alternative hypotheses can be integrated with customization of high-throughput cell-based assays , and with statistical methods for the optimal design of experiments [69] , so that every step of the Scientific Method can occur in a high-dimensional space .'], ['To model the biochemical behavior of the system , we used a system of ODEs to describe the evolution of the species concentrations over time .', 'Letting s denote the species and e denote which experiment or treatment condition is applied , the change of the transient concentration xs , e over time can be represented in ODE form as', 'ddtxs , e ( t , k ) =fs , e ( x ( t , k ) , i ( t ) , k )', '( 1 )', 'where t is the transient time point; x ( t , k ) is the set of species xs , e ( t , k ) that evolve over time , governed by the equations of the system; k is the parameter vector including reaction rates and initial concentrations; fs , e is an algebraic function derived from the biochemical reactions in the system , which may involve any parameters , species , inputs , or variable curves in the system; i ( t ) is the set of input species is , e ( t ) determined by predefined time-evolved functions rather than by the fs , e equations .', 'The species xs , e ( t , k ) are experiment-dependent because the concentration trajectory of a species can be affected by altered experimental conditions , addition of inhibitors , etc .', 'Although the inputs are not governed by the equations of the biochemical reactions , the inputs are also experiment-dependent to allow different conditions to cause different levels of inputs .', 'For H0 , x has six species [PDK1] , [PIP3:PDK1m] , [Aktc] , [PIP3:Aktm] , [Aktp308m] , and [Aktp308c] .', 'PIP3 is the only element of i .', 'Because PIP3 is an input curve and not simulated by ODEs , we cannot simulate mass action binding , with depletion of unbound PIP3 as PIP3 forms a complex with PDK1m or Aktm .', 'Instead , PIP3 appears as a simple multiplier in the equations describing the recruitment of PDK1 or Akt by PIP3 ( Table 1 , #1 and #3 ) .', 'The six ODEs for H0 are as follows:', 'd[PDK1]dt=−k1[PIP3][PDK1]+k2[PIP3:PDK1m]', '( 2 )', 'd[PIP3:PDK1m]dt=k1[PIP3][PDK1]−k2[PIP3:PDK1m]', '( 3 )', 'd[Aktc]dt=−k3[PIP3][Aktc]+k7[Phosphatase][Aktp308c]k8+[Aktp308c]', '( 4 )', 'd[PIP3:Aktm]dt=k3[PIP3][Aktc]−k4[PIP3:PDK1m][PIP3:Aktm]k5+[PIP3:Aktm]', '( 5 )', 'd[Aktp308m]dt=k4[PIP3:PDK1m][PIP3:Aktm]k5+[PIP3:Aktm]−k6[Aktp308m]', '( 6 )', 'd[Aktp308c]dt=k6[Aktp308m]−k7[Phosphatase][Aktp308c]k8+[Aktp308c]', '( 7 ) Models were implemented and simulated using Matlab ( v . 7 . 6 , The MathWorks , MA ) with SBToolbox2 toolkit [70] and its SBPD extension package .', 'In silico simulations used default settings for the ODE solver .', 'Models were originally implemented in TextBC format ( as required by SBToolbox2 ) , which contains model states , model variables and model reactions information .', 'SBML-formatted models are available in S1 Protocol .', 'Whenever applicable , published data and kinetic rate constants [10 , 31] were adopted to ensure that the model would not violate biological knowledge .', 'The remaining parameters were fitted to observed data in the process of model calibration .', 'The parameter estimation process can be expressed as a sum-of-squared-error ( SSE ) optimization problem:', 'mink{SSE}≡mink{∑s∑e∑tws , e⋅ ( xs , edata ( t ) −xs , e ( t , k ) ) 2}', '( 8 )', 'where xs , edata ( t ) is the experimental measure of species s , experiment e , time t , where ws , e=nt∑t ( xs , edata ( t ) ) 2 is a time-independent inverted mean-square weight , calculated based on the observed data , and nt is the number of time points [71] .', 'Eq ( 8 ) describes the objective function for a variety of optimization methods [70 , 72 , 73] applied to any of our models , not just H0 .', 'Model fitting for the H0 model used 200 initial starts [74 , 75] for global search ( Particle Swarm Optimization ) , followed by local search ( Nelder-Mead ) .', 'The algorithms were run using SBToolbox2 [70] with default settings .', 'Each non-canonical deviation consists of a pseudo-species ( a red oval in Fig 3 ) , and a pseudo-reaction ( a red dashed arrow in Fig 3 ) .', 'Each pseudo-species is defined by a linear spline , which represents the strength of the unknown effect ( literally , the abundance of the pseudo-species ) at each point in time .', 'A linear spline is a series of line segments connecting adjacent points called spline knots .', 'We parameterized each spline using 7 spline knots representing the strength of the non-canonical effect at time t = 0 , 2 , 5 , 10 , 30 , 60 , and 120min .', 'The spline knots are unknown variables , meaning that each pseudo-species starts out having unknown time-course .', 'A pseudo-reaction connects each pseudo-species to the rest of the pathway .', 'Like any biochemical reaction , a pseudo-reaction is described using the ordinary differential equations ( ODEs ) of the model .', 'The pseudo-species appear in the ODEs as input species , similar to PIP3 .', 'In mathematical terms , an input species appears in the right-hand side of the ODEs but not the left hand side .', 'In practical terms , an input species can drive biochemical reactions in the ODEs by serving as a reagent , but it is not governed by the ODEs , meaning the evolution of its level over time is not defined by solving the ODEs .', 'The spline knots are unknown variables , so they are treated like any other unknown parameter during the model fitting process .', 'Therefore the spline curve gets adjusted and optimized by the parameter estimation phase , so as to maximize the fit of the model to the data .', 'In other words , the parameter optimization process serves to estimate how much of the non-canonical effect would occur at each timepoint , in order to allow the overall model ( including the non-canonical effect ) to agree with the data .', 'Each alternative model was constructed by adding a non-canonical deviation ( S1 Text ) to the canonical pathway model H0 ( Eqs 1–7 ) .', 'Each non-canonical deviation ( a red addition in Fig 3 ) consists of a pseudo-species and a pseudo-reaction .', 'The pseudo-species are defined by splines , as described above , and the pseudo-reactions are described below as alterations to f , the set of ODEs in the original H0 model .', 'Although the splines v of the pseudo-species appear in the ODEs as “input species” and are not governed by the differential equations , they are still functions of the parameter vector k because they are defined by the spline knots , which are variables in the parameter vector k .', 'Therefore , the alternative models with non-canonical deviations can be represented with the following equation instead of Eq ( 1 ) :', 'ddtxs , e ( t , k ) =fs , e ( x ( t , k ) , i ( t ) , v ( t , k ) , k )', '( 9 )', 'where v ( t , k ) is the linear spline for the pseudo-species of the non-canonical deviation , and the differential equations f are modified as follows .', 'The equations for Model M1 are the same as for Model H0 except with Eqs ( 10 and 11 ) in place of Eqs ( 4 and 5 ) .', 'M2 is also based on model H0 but with Eqs ( 12 and 13 ) instead of Eqs ( 4 and 5 ) .', 'Model M3 has a change in species as well as equations .', 'M3 is the same as H0 except for Eqs ( 6 and 7 ) being replaced by Eqs ( 14–16 ) , and species Aktp308m being replaced by a pair of species Aktp308mf ( representing membrane Aktp308 that is free to dissociate from the membrane ) and Aktp308mt ( membrane Aktp308 that is trapped at the membrane ) .', 'Model M4 is based on model H0 except with additional Eqs ( 17 and 18 ) .', 'Model M5 is based on model H0 but with Eqs ( 19 and 20 ) instead of Eqs ( 5 and 6 ) .', 'd[PIP3:Aktm]dt=k3[PIP3][Aktc]−k4[PIP3:PDK1m][Phosphorylation][PIP3:Aktm]k5+[PIP3:Aktm]', '( 19 )', 'd[Aktp308m]dt=k4[PIP3:PDK1m][Phosphorylation][PIP3:Aktm]k5+[PIP3:Aktm]−k6[Aktp308m]', '( 20 )', 'In additional to the rate parameters k1-k8 from the canonical model ( Table 1 ) , the alternative models ( M1-M5 ) include new parameters k9-k13 , the values of which must be estimated during the parameter estimation process .', 'The Phosphatase level was held constant for all models except M5 .', 'Each model is fit to the data using parameter estimation .', 'As shown in the flowchart of Fig 3F , parameter estimation occurs by performing many iterations of simulating the model ( Box 1 ) while changing the values of the variable parameters .', 'In that way , parameter estimation for the alternative models is exactly similar to parameter estimation for H0 , although the param
['Introduction', 'Results', 'Discussion', 'Methods']
['The dynamic behaviors of signaling pathways can provide clues to pathway mechanisms .', 'In cancer cells , excessive phosphorylation and activation of the Akt pathway is responsible for cell survival advantages .', 'In normal cells , serum stimulation causes brief peaks of extremely high Akt phosphorylation before reaching a moderate steady-state .', 'Previous modeling assumed this peak and decline behavior ( i . e . , “overshoot” ) was due to receptor internalization .', 'In this work , we modeled the dynamics of the overshoot as a tool for gaining insight into Akt pathway function .', 'We built an ordinary differential equation ( ODE ) model describing pathway activation immediately upstream of Akt phosphorylation at Thr308 ( Aktp308 ) .', 'The model was fit to experimental measurements of Aktp308 , total Akt , and phosphatidylinositol ( 3 , 4 , 5 ) -trisphosphate ( PIP3 ) , from mouse embryonic fibroblasts with serum stimulation .', 'The canonical Akt activation model ( the null hypothesis ) was unable to recapitulate the observed delay between the peak of PIP3 ( at 2 minutes ) , and the peak of Aktp308 ( at 30–60 minutes ) .', 'From this we conclude that the peak and decline behavior of Aktp308 is not caused by PIP3 dynamics .', 'Models for alternative hypotheses were constructed by allowing an arbitrary dynamic curve to perturb each of 5 steps of the pathway .', 'All 5 of the alternative models could reproduce the observed delay .', 'To distinguish among the alternatives , simulations suggested which species and timepoints would show strong differences .', 'Time-series experiments with membrane fractionation and PI3K inhibition were performed , and incompatible hypotheses were excluded .', 'We conclude that the peak and decline behavior of Aktp308 is caused by a non-canonical effect that retains Akt at the membrane , and not by receptor internalization .', 'Furthermore , we provide a novel spline-based method for simulating the network implications of an unknown effect , and we demonstrate a process of hypothesis management for guiding efficient experiments .']
['Influential pathways of cell signalling ( such as PI3K/Akt ) are routinely communicated using simple textbook-like diagrams that show only the most widely-accepted steps of the pathway .', 'At the same time , there are countless other molecular influences relevant to each pathway , documented in the published literature , and more are being published every week .', 'It should perhaps come as little surprise that during a routine observation of the Akt activation pathway , a simulation of the canonical model was mathematically incompatible with our observed dynamics .', 'To progress beyond the standard , simplified model without testing an unreasonable number of molecular candidates individually , we employed computational modeling to analyze the dynamics of pathway activation .', 'We asked when and where a non-canonical deviation could occur , relative to the canonical pathway .', 'We used the timing of downstream activation to solve for the possible times of upstream initiation .', 'By categorizing unknown effects by their dynamics , we were able to prune away implausible hypotheses using an efficient number of in vitro experiments .', 'At the end we had a single plausible explanation for non-canonical Akt activation in our cells , and we confirmed experimentally that Akt is retained at the membrane after PIP3 is no longer present .']
[]