article: Proteasomal protein degradation is a key determinant of protein half-life and hence of cellular processes ranging from basic metabolism to a host of immunological processes. Despite its importance the mechanisms regulating proteasome activity are only incompletely understood. Here we use an iterative and tightly integrated experimental and modelling approach to develop, explore and validate mechanistic models of proteasomal peptide-hydrolysis dynamics. The 20S proteasome is a dynamic enzyme and its activity varies over time because of interactions between substrates and products and the proteolytic and regulatory sites; the locations of these sites and the interactions between them are predicted by the model, and experimentally supported. The analysis suggests that the rate-limiting step of hydrolysis is the transport of the substrates into the proteasome. The transport efficiency varies between human standard- and immuno-proteasomes thereby impinging upon total degradation rate and substrate cleavage-site usage.
  Cells can be usefully thought of as communities of molecular machines that work in concert to maintain cellular function (Nurse, 2003). Metabolism, gene expression, protein production and degradation, energy production, DNA replication, cell division all have their own repertoire of associated proteins and macromolecular assemblies, and are controlled by complex and highly dynamic signalling and regulatory networks; this picture is complemented by a slew of structural proteins that provide cell walls, nuclear membranes, endoplasmic reticulum and other structures of the eukaryotic cell. But while advances in imaging and microscopy technology have allowed us to glean some insights into the structures and even dynamics at the molecular level (Liepe et al. , 2012), much of our knowledge about the function of for example, protein production at ribosomes, or degradation at proteasomes, is based on indirect observations, or on data collected at some experimentally convenient equilibrium state. Increasingly time-resolved data can also be collected, and this type of data typically provides more detailed insights into the molecular mechanisms at work. To interpret such data, mathematical modelling coupled to state-of-the-art statistical inference thus becomes a necessity (May 2004; Buchholz et al. , 2013; Gerlach et al. , 2013) to make sense of data and design better, more discriminatory experiments. Here we apply a modelling approach to a detailed mechanistic analysis of proteasomes, which are at the core of the ubiquitin-proteasome system and responsible for the destruction of the majority of the cytoplasmic proteins. Proteins are usually tagged by E1-E2-E3-E4 enzyme-mediated poly-ubiquitination, carried into the proteasome proteolytic chamber where they are fragmented and, ultimately, expulsed (Schwartz and Ciechanover, 2009). Proteins, and especially oxidised proteins—that is, those that reflect the presence of cellular stress conditions—can also be degraded by 20S proteasomes without prior poly-ubiquitination (Aiken et al. , 2011; Pickering and Davies, 2012; Ben-Nissan and Sharon, 2014; Höhn and Grune, 2014). In addition to stress response, the 20S proteasomes present in cells (Fabre et al. , 2015) have been shown to regulate the abundance of a host of signalling molecules involved in cell cycle progression, cellular growth control and oncogenesis (Ben-Nissan and Sharon, 2014). Additionally, 20S proteasomes form the core of the structure of 26S proteasomes and we cannot expect to understand the latter if we do not understand the mechanisms that determine the activity of the former. Poly-ubiquitin tagging (when it occurs), transport and peptide-bond hydrolysis regulate protein half-life and thereby affect the majority of metabolic processes in the cell. Despite considerable biochemical and structural efforts, the mechanisms of proteasomal action, in particular the causes and extent of differences in substrate-specific proteolysis between different proteasome isoforms, are only incompletely understood. In this study we combine carefully designed experimental assays with detailed modelling to shed light on these mechanisms. There is a dearth of mechanistic analyses of proteasome functions, and the complexity of polypeptide hydrolysis has precluded detailed analysis (Liepe et al. , 2014a). The most direct insights come from carefully chosen short fluorogenic peptides, which have a single proteasome-catalyzed cleavage site, which in turn allows us to follow the kinetics in real time. They have also been used (exclusively) to measure the proteasomal activity in cellulo (e. g. , Kapeta et al. , 2010; Chondrogianni et al. , 2015; Peters et al. , 2015; Pickering et al. , 2015), despite the fact that they lack the complexity of ‘real’ proteins and they do not recapitulate proteasome proteolytic activities towards polypeptide chains (Mishto et al. , 2014). Nevertheless, they provide arguably the most promising class of leads of proteasomal inhibitors (Bellavista et al. , 2013; Kisselev and Groettrup, 2014). Furthermore, kinetic and structural analyses rely on their structural and biophysical characteristics (Gaczynska et al. , 1993; Kisselev et al. , 2002,2003,2006; Osmulski et al. , 2009); we, too, use them here for our analysis, which is based on a representative set of such peptides. As we show below the insights gained from the short fluorogenic peptides are borne out by further analysis of polypeptides. The mechanistic analyses performed here also incorporate known structural features into a mathematical model of proteasome action. The 20S proteasome consists of four stacked seven membered rings (denoted by α7β7β7α7). These rings form three interconnected cavities, including a pair of antechambers through which substrates are passed before reaching the central catalytic chamber. Antechamber walls are not merely structural, but interact actively with the substrates by altering their properties and keeping them accessible for hydrolysis (Ruschak et al. , 2010). These proteasome chambers can also store two or more proteins in order to enable continuous degradation (Hutschenreiter et al. , 2004; Sharon et al. , 2006). The central chamber contains six subunits (two β1, β2 and β5 subunits) that catalyse the peptide-bond hydrolysis and peptide splicing after binding of the polypeptide substrates nearby their active N-terminal Thr (Vigneron et al. , 2004; Borissenko and Groll, 2007; Mishto et al. , 2012). 20S proteasomes modify their conformations and thus in turn their activity upon peptide-bond hydrolysis (Osmulski et al. , 2009; Ruschak and Kay, 2012), binding of regulatory complexes such as 11S or 19S to the proteasome α rings (leading to the formation of PA28-capped proteasomes and 26S proteasomes, respectively) (Dick et al. , 1996; Emmerich et al. , 2000; Köhler et al. , 2001; Ruschak and Kay, 2012; Raule et al. , 2014), and activation of non-catalytic modifier sites (Schmidtke et al. , 2000; Kisselev et al. , 2002,2003) whose location in the proteasome remain unknown (Liepe et al. , 2014a). In mammals different 20S proteasome isoforms exist, which carry different catalytic β subunits. Upon an inflammatory stimulus such as IFN-γ the catalytic standard β1, β2 and β5 subunits peculiar of the standard proteasome (s-proteasome) are replaced by the immuno-subunits β1i, β2i and β5i in the newly synthesized immunoproteasome (i-proteasome). Proteasome isoforms degrade substrates with different rates, but generate the same peptide pool (Mishto et al. , 2014). And while there exist differences between substrates, the 20S i-proteasome has been shown often to have a higher polypeptide degradation rate than its standard counterpart (Bellavista et al. , 2013). Why s- or i-proteasomes should degrade specific substrates more rapidly is a matter of ongoing debate. Different steps of the proteolysis process may be responsible for isoform specific kinetics: transport of the substrate through the gate and the antechambers; binding to substrate binding sites and ensuing peptide-bond hydrolysis (Huber et al. , 2012; Arciniega et al. , 2014); and the release from the substrate binding sites and finally from the proteasome gate. The question is also of direct medical relevance: different degradation kinetics of specific substrates by s- and i-proteasomes, or i-proteasome carrying genetic polymorphisms/mutations, have been suggested to be involved in a variety of pathologies (Basler et al. , 2013; Bellavista et al. , 2013). Furthermore, i-proteasome deficient mice show altered proteasome-dependent kinetics of pathogen epitope generation (Basler et al. , 2013). Such different kinetics could, for example, lead to the remodelling of immunodominance of viral epitopes (Zanker et al. , 2013) or to the lack of CD8+ T cell-mediated response towards specific viral antigens (Deol et al. , 2007). As we will show below, our framework allows us to start from simple models of proteasome function, and identify which mechanistic shortcomings they exhibit. We focus on the dynamics of the digestion of a set of exemplar peptides, and show that this process is regulated carefully and by three distinct mechanisms, that act in concert to regulate proteasomal activity. Here considering different mechanisms, coupled to state-of-the art statistical model selection techniques (Kirk et al. , 2013), allows us to elucidate steps in the proteasomal dynamics that cannot be probed directly through experiments. Model selection naturally extends the conventional hypothesis testing approach, and by accounting for uncertainty fairly and correctly, can be applied to systems with many unknown kinetic parameters (which are, in any case, inferred together with their respective uncertainties) (Liepe et al. , 2013; Babtie et al. , 2014).
  Using tightly integrated experimental and computational modelling analyses, recent advances in our understanding of proteasome structure and function, and previous attempts at modelling proteasome dynamics (Liepe et al. , 2014a), we elaborate the first comprehensive mathematical model that is able to describe the regulatory components of 20S proteasomes; and the complex interactions between substrate/product transport and substrate hydrolysis over time for representative peptides. Modelling helps us to understand how the 20S proteasome catalyzes the degradation of specific proteins in cells (Pickering and Davies, 2012; Ben-Nissan and Sharon, 2014; Höhn and Grune, 2014; Fabre et al. , 2015); it also forms the basis for future studies of the main active forms of proteasome in cells, for example, when it is bound to the regulatory complexes 19S and PA28 (Fabre et al. , 2015). Indeed, in 19S or PA28 single-capped proteasomes all steps considered in our model are still present, including the regulation of the gate by non-catalytic modifier site (s) and the gate binding affinity since one of the proteasome gates is not bound to the regulatory complexes and will therefore be regulated as described here. We observe similar proteasome dynamics over time when using purified 20S proteasome as well as protein homogenates, where 20S proteasomes are surrounded by several regulatory molecules. It suggests that with the present model we describe a core of proteasome features that could reflect proteasome dynamics also in more complex scenarios such as might exist inside the cellular environment. Our model therefore also provides an interpretative framework for general studies of proteasome action, keeping in mind that substrate-specific effects need to be accounted for (Mishto et al. , 2014). Simple (e. g. , MM-type) models, which have been largely used in analysing proteasome functionality so far, are clearly not capable of describing proteasome dynamics; this is becoming apparent since time-course data is much less forgiving and harder to fit to than for example, dose-response curves. The 20S proteasome is a dynamic enzyme and its catalytic activity varies over time in a way that can only be understood if the structural characteristics of the proteasome are taken into account. Kinetics change over longer time scales than the structural dynamics described by Osmulski et al. (2009) and they are influenced by the interactions of substrates and products with both proteolytic and regulatory sites. Our analysis suggests that the cavity of 20S proteasomes is progressively filled by peptides that can further facilitate other molecules to enter the chambers until the effective cavity volume has been almost completely filled. In fact, the estimated numbers of substrate and product molecules (C) that are inside the proteasome cavity are close to the possible maxima (Box 2). Peptide accumulation over time leads to activation of non-catalytic modifier sites, which our analysis shows to be located inside the proteasome cavity. 10. 7554/eLife. 07545. 025Box 2. Filling kinetics of the proteasome inner cavity over time. A key part of the peptide transport dynamics is the maximum capacity (C) of the two proteasome ante-chambers and the main chamber. C describes the maximal number of molecules of a given substrate can be allocated inside the proteasome chamber at the same time. C is dependent on the volume of the proteasome chamber and on the volume of the substrate molecules. For this reason C is expected to be substrate specific. The model estimates of C for the three tested substrates are in agreement with values predicted by computing the maximal number of substrates that could be located inside the proteasome chamber. The estimated volumes for Suc-LLVY-MCA, Z-LLE-MCA and Bz-VGR-MCA are approx. 936 A3,741 A3 and 606 A3, respectively (with an MCA group of 200 A3). This results in a ratio of 1. 25 for Suc-LLVY-MCA: Z-LLE-MCA (model estimate: 1. 24), 1. 36 for Suc-LLVY-MCA: Bz-VGR-MCA (model estimate: 1. 46) and 1. 08 for Z-LLE-MCA: Bz-VGR-MCA (model estimate: 1. 12). The exact number of molecules inside the chamber depends on the maximum possible density, but is approximately estimated to be 250,310 and 337 for Suc-LLVY-MCA, Z-LLE-MCA and Bz- VGR-MCA, respectively, which is in the same order of magnitude as the model estimates (200,249 and 293). Note, computed values are the maximal possible amount of molecules that could be packed inside the chamber. However, under physiological conditions the actual number of molecules that can be filled in the proteasome cavity has to be smaller, which is reflected in our model estimates. DOI: http: //dx. doi. org/10. 7554/eLife. 07545. 025 Substrate transport through the gate and cavities must therefore be a key factor in the regulation of the proteolytic process, despite the fact that it had been largely neglected so far in the quantitative analysis of proteasome dynamics (Liepe et al. , 2014a). The integration of data into our mathematical model clearly shows that peptide transport is the major rate limiting step in the degradation of short fluorogenic substrates: all models/hypotheses that do not account for this are soundly rejected. Hydrolysis is only rate limiting for the degradation of Z-LLE-MCA. We stress that for the short fluorogenic substrates the rate limiting steps are conserved across the different proteasomes (mouse proteasome and human s- and i-proteasomes). This in turn suggests that the interspecies homology of the proteasome structure results in preserving transport characteristics, although subtle structural variations can quantitatively modify the transport, as observed by comparing s- and i-proteasomes. Variation of the transport efficiency typically leads to quantitative changes in the degradation rates of short fluorogenic substrates. For more complex substrates, for example, polypeptides, transport variation could change substrate cleavage-site usage and thus result in the generation of specific peptide products. If these peptides are MHC class I-restricted epitopes, substantial variations in their amount could strongly affect the cell-mediated immune response. Indeed, as we have recently shown, significant decreases of antigenic peptide amount produced by proteasome could lead to a presentation onto the MHC class I molecules that is so diminished to be not able to trigger CD8+ T cell activation (Mishto et al. , 2014). According to our model, modifications of the proteasome gate upon PA28 binding will for instance affect the substrate cleavage-site usage and lead to the observed alterations of MHC class I-restricted epitope repertoire (Cascio, 2014). Differences in gate and inner channel features are in part responsible for the different degradation rates of the short fluorogenic peptides by human s- and i-proteasomes although they diverge also in the activity of the catalytic subunits themselves. Differences in the gate between the two proteasome isoforms are supported also by Fabre et al. (2015), who identified preferential binding between the α subunits of s- and i-proteasomes and specific regulatory complexes. Such differences can become dominant driving forces. For instance, the degradation of the substrate Bz-VGR-MCA is not influenced by differences in the active site-related parameters—as is correctly predicted by our model. This also agrees with comparisons of the crystallographic structures of the mouse s- and i-proteasomes, which identify only minor variations between the β2 and β2i catalytic pockets (Huber et al. , 2012). By contrast, the catalytic sites of s-proteasome have stronger binding affinity for the substrate Z-LLE-MCA than the i-proteasome, which could be explained by the structural differences between β1 and β1i catalytic pockets described by Huber et al. (2012). For the degradation of the substrate Suc-LLVY-MCA our model predicts that the i-proteasome has higher substrate-binding affinity, higher hydrolysis rate, and lower cooperativity than the s-proteasome. This prediction is corroborated by the recent study of Arciniega et al. (2014), who showed that the β5i pocket is more prone to bind and process substrates than the β5 pocket; however, only s-proteasome is able to modify its conformation upon substrate binding to the β5 subunit (Arciniega et al. , 2014), which is also correctly captured by our model as increased cooperativity (ni). The outcomes of our integrative analysis suggest that the i-proteasome also has a higher affinity for the enhancing non-catalytic modifier site activated by all three substrates; given that our model consistently predicts this site to be located at the inner surface, it should now become possible to determine the precise location of the enhancer site using structural techniques. Our model is a necessary and non-trivial step towards understanding protein degradation by proteasomes. A full-length protein is a much more complicated substrate and it can interact with the different regulatory and catalytic sites in a myriad of ways. In any such analysis, however, the model developed here can aid the experimental set-up (Liepe et al. , 2013; Sunnaker et al. , 2013) and the interpretation of experimental data. What is already clear is that the complex interactions between a suitable complicated substrate and the proteasomal machinery will substantially shape differences in degradation rates between proteins; as the protein degradation rate is known to influence for example, noise in signal transduction, a way to understand and rationally interfere with this process is of obvious importance.
  The final model shown in Figure 2J contains a set of ordinary differential equations as follows: dSoutdt=−SoutG1kon+[G1Sout]koff+[G2S]transportout−hSouthIfreeIon+h[IS]IoffdG1dt=− (Sout+Pout) G1kon+ ([G1Sout]+[G1Pout]) (koff+transportin) dPoutdt=−PoutG1kon+[G1Pout]koff+[G2P]transportout−hPouthIfreeIon+h[IP]Ioffd[G1Sout]dt=SoutG1kon−[G1Sout] (koff+transportin) d[G1Pout]dt=PoutG1kon−[G1Pout] (koff+transportin) dSdt=[G1Sout]transportin−τSG2E0−vhydr−RonSEregE0+Roff[EregS]dPdt=[G1Pout]transportin−τPG2E0+vhydr−RonPEregE0+Roff[EregP]d[G2S]dt=τG2SE0−[G2S]transportoutd[G2]dt=τG2 (S+P) E0− ([G2S]+[G2P]) transportoutd[G2P]dt=τG2PE0−[G2P]transportoutd[Ereg]dt=−RonEregE0 (S+P) +Roff ([EregS]+[EregP]) d[EregS]dt=RonEregE0S−Roff[EregS]d[EregP]dt=RonEregE0P−Roff[EregP]dIfreedt=− (South+Pouth) IfreeIon+Ioff ([IS]+[IP]) d[IS]dt=SouthIfreeIon+Ioff[IS]d[IP]dt=PouthIfreeIon+Ioff[IP], withtransportin=vin1+Xenh[EregS]+[EregP]E01+Yinh[IS]+[IP]I0tanh (E0C−S−P) transportout=vout1+Xenh[EregS]+[EregP]E01+Yinh[IS]+[IP]I0vhydr=nakpE0SxKaS (1+βSniαKiS+βPniαKiP) x=1+SnaKaS+SniKiS+PnaKaP+PniKiP+Sna+niαKaSKiS+Pna+niαKaPKiP+SnaPniαKaSKiP+SniPnaαKiSKaP. A full list of parameters and model species is given in Figure 2—source data 2,3. Details about the ‘Materials and methods’, mathematical analysis and other tested models can be found in Supplementary file 1. The model is also provided in SBML format (generated with the SBML editor from MINRES Technologies) in Supplementary file 1.
summary: Cells have to be able to reliably destroy or remove molecules from their interior that they no longer need. Structures called proteasomes play a central part in this complex process by cutting up and digesting proteins. Mammals have several different types of proteasomes, each made up of several protein ‘subunits’. For example, when a cell experiences inflammation some proteasomes change some of their subunits and form an immuno-proteasome. These immuno-proteasomes tend to break down proteins more quickly than ‘standard’ proteasomes, but it was not clear how they are able to do so. Liepe et al. have now combined experiments and mathematical modelling to construct a detailed model of proteasome activity. The model shows that protein transport into and out of the proteasome chamber are the steps that limit how quickly the proteasomes can break down proteins. Furthermore, these transport processes are also to a large extent responsible for the different rates at which standard and immuno-proteasomes process proteins. Liepe et al. were also able to confirm the existence of regulatory sites within the proteasome, and describe how these are arranged. Problems that alter the rate at which proteasomes break down proteins have been linked to tumors and neurological and autoimmune diseases. Liepe et al.' s model opens up the ability to study how the proteasome' s activity is affected by drugs and therefore makes it easier to investigate ways of interfering with this activity for therapeutic purposes.
keywords: computational and systems biology
title: Quantitative time-resolved analysis reveals intricate, differential regulation of standard- and immuno-proteasomes