Unraveling Apoptosis Signalling using Linear Control Methods: Linking the Loop Gain to Reverting the Decision to Undergo Apoptosis

Transcription

1 Preprints of the 9th International Symposium on Advanced Control of Chemical Processes The International Federation of Automatic Control TuPoster.3 Unraveling Apoptosis Signalling using Linear Control Methods: Linking the Loop Gain to Reverting the Decision to Undergo Apoptosis Monica Schliemann-Bullinger Mark C. Readman Dimitrios Kalamatianos Rolf Findeisen Eric Bullinger Laboratory for Systems Theory and Automatic Control, Otto-von-Guericke University, Magdeburg, Germany Stockport College, Stockport, United Kingdom and School of Mathematics, The University of Manchester, Manchester, United Kingdom Division of Developmental Biology, Biomedical Research Foundation of the Academy of Athens, Athens, Greece Abstract: Apoptosis programmed cell death is a key component in cell renewal, i.e. for removing damaged cells or those not any more required without harming their environment. Understanding apoptosis is challenging since the involved signalling network is complex and intertwined. We use linear control theory to shine light on the complex signalling mechanisms and interactions. To this end we linearise different models around the steady state corresponding to the unstimulated case and use linear control analysis to uncover and identify a decentralised control scheme with two plant and two controller modules that are sparsely interconnected. While this separation on a first view is artificial, it allows for using the small gain theorem to analyse stability and to give insight into how inhibitors naturally modulate cell death. This sheds light onto the role of positive and negative feedback present in this signalling pathway. Sensitivity analysis reveals these most influential parameters on the time of death. To illustrate the analysis, we examine in simulations to which extent apoptosis can be reversed once initiated. Keywords: apoptosis, linear control theory, robustness, perturbation analysis, biochemical systems.. INTRODUCTION An essential intracellular mechanism for maintaining homoeostasis in living organisms is apoptosis, also called programmed cell death. Apoptosis is a key regulator for replacing unused, old or damaged cell and counterbalances proliferation. While apoptosis signalling pathways are present in most cells, the essential property of a healthy organisms is the right balance between life and death. For example, in dementia, apoptosis is triggered prematurely and here the probability of apoptosis should be restored to the normal decision level time of the particular cell type [Kao et al., ]. Apoptosis is downregulated in cancer cells, here one would like to trigger apoptosis more frequently [Hanahan and Weinberg, ]. Apoptosis signalling can be induced via the intrinsic pathway by intracellular signals, e.g. DNA damage or oxidative stress [Eißing et al., ]. Alternatively, extracellular cy- This work was in part supported by the BMBF-funded E:Bio- Project InTraSig (Inhibiting Trans-Signalling). tokines such as FasL, TRAIL or TNF trigger the extrinsic pathway [Rehm et al., 6]. In both cases, caspase inhibitors regulate this process. We propose to exploit linear control theory at the life steady state to clarify the antagonistic relationships between negative and positive feedback loops. These analysis approaches enables us to suggest targets and strategies for modulating and controlling apoptosis, e.g. reversing apoptosis once erroneously activated. By now, several mathematical models of apoptosis have been published [Huber, Bullinger and Rehm, 9]. We consider three of these with different complexity, all consisting of ordinary differential equations having reactions with mass action kinetics based on [Eißing et al., ; Albeck et al., 8; Schliemann et al., ] to underpin the presence of basic control mechanism. We extend the modularisation of the system as proposed in [Readman et al., 3], which considers the feedback loops as an interconnection of plant and controller, revealing the control role of the inhibitors. Here, we analyse the feedback loop of caspases, in particular from a loop gain Copyright 5 IFAC 955

2 and sensitivity analysis perspective leading to targets for the reversal of apoptosis. The rest of this paper is organized as follows. Section describes the apoptosis signalling and its modularisation, setting the basis to Section 3 that performs a loop gain analysis. A parametric sensitivity analysis is performed in Section while Section 5 links loop-gain and sensitivity analysis results to derive the most influential factors to influence the time between apoptosis decision and irreversible decision for death counteracting apoptosis.. APOTOSIS SIGNALLING This section starts with an overview of the key biological players of apoptosis signalling before presenting the modularisation approach at the basis of the current study.. Biological Overview The death signal is initiated either at a death receptor (extrinsic apoptosis pathway) or by an intracellular signal (intrinsic case). Common to both intrinsic and extrinsic apoptotic pathways is a positive feedback loop, where the caspase proteins (cysteine-dependent aspartate-directed proteases) activate each other. First, initiator caspases, e.g. Caspase-8, are activated by the intrinsic or extrinsic stimulus. Then, initiator caspases activate effector caspases, such as Caspase-3. These in turn activate more initiator caspases. Activation of a sufficient amount of effector caspases leads to the irreversible start of the cellular suicide programme. In type I cells, the feedback is able to activate enough effector caspases for starting the death programme. Type II cells however, require an additional activation, where the initiator caspases trigger the release of pro-apoptotic proteins such as cytochrome-c from the mitochondria.. Modules of Apoptosis Signalling The present manuscript analyses three published signalling models of apoptosis. The core reaction of apoptosis induction are encoded in the Eißing model [Eißing et al., ], which contains a positive feedback loop of activator caspases (Caspase 8) and effector caspases (Caspases 3). These are stabilised by two specific inhibitors acting on them, namely BAR (Bifunctional apoptosis inhibitor) on Caspase 8 and (X-linked inhibitor of apoptosis protein) on Caspases 3, see Figure and [Readman et al., 3, Supplement] for the details. Significantly larger are the other two models: Albeck model [Albeck et al., 8] and Schliemann model [Schliemann et al., ]. Both include the signalling pathway from the death receptor to the caspase loop as well as an extra caspase in the positive feedback loop, i.e. Caspase 6. While the Albeck model also includes the mitochondrial pathway acting as death signal amplifier for type II cells, the Schliemann model contains the NF-κB pathway responsible for the expression of inhibitors of apoptosis. Details on the models can be found in the original references as well as in [Readman et al., 3] and in particular in its Supplemental Information. All three models contain a positive feedback loop, which helps in quickly activating after a possibly long Table. Properties of the considered apoptosis models. model name Albeck Eißing Schliemann components unit amount amount µm steady-states 3 3 life steady-state unstable stable stable compartments 3 reversible reactions irrevers. reactions kinetic parameters type I / II II core only I intrinsic / extrinsic extrinsic core only extrinsic source Trigger input Death signal Plant-Controller u = y Plant pcasp6 Casp6 Casp3 Plant pcasp3 Casp3 Albeck et al., 8 pcasp8 Casp8 Casp8 Plant-Controller IKKa y = u BAR Casp8 Casp3 Eißing et al., Controller 3 NF-κB Controller BAR BAR~Casp8 A- & FLIP mrna Controller ~Casp3 Schliemann et al., mrna Figure. Systems View on Apoptosis. The models can be structurally decomposed into five modules, two plant and three controller modules. Controller 3 is only present in the Schliemann model. All three models have for modules, as the Controller 3 module only present in the Schliemann model is almost constant. Grouping Plant with Controller and Plant with Controller results in an interconnection of two subsystems, where the interconnection is physically given by components of the caspase loop. The first subsystem receives the input signal triggering apoptosis (e.g. the extracellular ligand), while the second subsystem produces the effective apoptotic signal, in form of the effector caspase Caspase-3. delay, see Figure. Understanding the purpose of the apoptotic signalling pathway is not straight forward. We follow the modularisation proposed by Readman et al., 3, which splits the states in a way minimising the number of interconnections between the modules. This results in four modules, denoted Plant and and Controller and, see Figure. The fifth module, Controller 3, is only present in the Schliemann model. As its external input is practically zero in the neighbourhood of the life steady state, this module can well be approximated by a constant. Thus, all four models are partitioned into four modules. The three models consist of differential equations of the form ċ = Nv(c, u) Copyright 5 IFAC 956

3 where N is the constant stoichiometric matrix, v the vector of fluxes, c is the vector of non-negative concentrations and u the input signal. In the absence of an input, i.e. u =, the initial condition c is the steady-state where all caspases are inactive. This corresponds to normal cell behaviour and is called life steady-state. To better understand the regulation we linearise the models around the life steady-state c, in which the systems are in the absence of a death triggering signal. This leads for each model to a system of linear ordinary differential equations ẋ = Ax + Bu where x = c c and v(c, u) A = N c c=c, u=, B = N v(c, u) u c=c, u= For analysing the interconnected system of modules plants and controllers, the plant and controller modules with identical index were combined by first reordering the state vector of the full models, x, to x as follows: x (P) x = x (C) x (P), x (C) where x (P) are the states of Plant, x (C) are the states of Controller, etc., while x (i) are the states of the combination of the modules Plant i and Controller i. Then, [ ] [ ] x () x (P ) =, x () x (P ) = x (C) is the resulting partitioning studied here. x (C) As all reactions are assumed to follow the law of mass action, the Jacobian matrices can be easily computed symbolically. In particular, their entries are polynomial in the states and linear in the parameters. The discussed modularisation corresponds to breaking the positive feedback loop of the caspases as the initiator caspases belong to the Plant-Controller subsystem, while the effector caspases are part of Plant-Controller. The corresponding dynamical systems are Plant-Controller Plant-Controller ẋ = A x + B u, ẋ = A x + B u y = C x, y = C x + D u interconnected by u = y u = y. For all three models under study here, Plant-Controller has no feed-through term, in contrast to Plant-Controller which has a non-zero D. For the Eißing and Schliemann models, the resulting Plant-Controller systems are SISO, while in the Albeck case, the systems are MIMO: For Albeck, the systems are MIMO, Plant-Controller has 6 inputs and 3 outputs. This is due to the Albeck model containing intermediate components in the caspase activation loop, e.g. complexes such as Caspase-8 Procaspase-3. To get a better understanding of these models, the system parts not relevant for the input-output behaviour of a. subsystem were removed using balanced model reduction (performed with MATLAB s balancmr). For each Plant-Controller linearised at the life steady-state the Hankel singular values are calculated, see Figure. These show that the systems can be well approximated by low order models close to these..3 Transfer Functions of the Modules For the Eißing model, the transfer functions are approximately.8(s +.6) G (s) (s +.)(s +.5), G ().3 db.3(s +.73) G (s) (s +.)(s +.78), G (). db where G is the transfer function for Plant-Controller and G for Plant-Controller. For the Schliemann model,.377(s ) G (s) (s +.3)(s +.353),.36(s +.667) G (s) (s + 9.)(s +.67), such that G () 7. db and G () 3. db. For the Albeck model we obtain [ ].37 G (s) s +.37 ( et e T 3 + e T ) [ ].6 s (s +.379)(s +.879) (e + e 6 ) T [ ].96 s e T 5 d (s) with e i the i-th Euclidean basis vector in R 6, i.e. e ij = for i = j and otherwise, and d (s) = (s + 6.)(s +.379)(s +.879)(s +.37), 36(s + 6.6) G (s).6 6 where (s + 6.6)(s + 6.6) d (s) [ ].5(s + 6.6)(s + 6)(s + 6.6) + e 3 [ ] (s +.37)d (s) e s [ ] + e [ ] s e 3 [ ], s +.37 d (s) = (s + 7.5)(s + 6.7)(s + 6.6) (s )(s + 5.8). It is interesting to note that all Plant-Controller systems are asymptotically stable and strictly proper. In the following sections we will analyse the interconnection between the two subsystems from a loop gain perspective and relate Copyright 5 IFAC 957

4 x 5 6 (c) Albeck model Gain [db] Gain [db] Frequency [rad/min] Frequency [rad/min] (c) Albeck model. Gain [db] Frequency [rad/min]. Figure. Hankel Singular Values of the systems Plant- Controller (top) and Plant-Controller (bottom). this to the possibility of reversing apoptosis in the phase of delayed decision making. 3. H VIEW OF THE APOPTOSIS FEEDBACK LOOP A sufficient condition for stability of the closed-loop interconnection of Plant-Controller and Plant-Controller as in Figure is that both are asymptotically stable and the loop gain, i.e. the product of their maximal gain in the induced L sense, also known as H norm, is smaller than one. All three models analysed here have asymptotically stable Plant-Controller modules, see Section.3. The gain of the modules can be visualised in singular value plots, which shows the maximal singular value of the transfer function matrix evaluated along the frequency axis. Figure 3 shows the maximal singular values for Plant- Controller- and - as well as their product (sum in the logarithmic decibel scale). The analysis shows that the Eißing and Schliemann models are stable as both products are smaller than = db, though not with a very large margin. This proximity to the stability boundary is a key ingredient enabling delayed decision making [Trotta, Bullinger and Sepulchre, ]. The Albeck model has, however, a loop gain that is significantly larger than one, due to Plant-Controller- and no damping in Plant-Controller-, compared to the Schliemann model. Thus, the loop gain analysis is not informative for inferring the closed loop stability. Interesting is to note that in the two more detailed models, i.e. those by Albeck and Schliemann, the gain of Plant-Controller- is significantly larger than in Plant- Controller-. As the trigger signal enters the first and the latter is producing the overall output signals, the caspase loop has the seemingly counter-intuitive property of first amplifying the trigger signal before damping it. These loop gain analyses were performed at the life steady state. The loop gain can also be calculated along a dynamic trajectory, as is down in Section 5. Figure 3. Maximal singular value plots of the systems Plant-Controller (blue) and Plant-Controller (green), as well as their sum (red), which is an upper bound for the loop gain.. SENSITIVITY ANALYSIS OF THE LOOP GAIN To uncover the most important parameters for apoptosis analysis of the relative disease, we perturbed each parameter one at the time by the factor. and we calculated the relative sensitivity [Neumann et al., ; Shoemaker and Doyle, 8] of the loop gain, see Figure and Table. The computational effort is higher than sensitivity analyses of steady states, but significantly lower than the sensitivity of the time of death, which requires solving the nonlinear and stiff systems of differential equations. For the Eißing model, the most sensitive parameters are the mutual caspase activation rates, as well as the association, dissociation and degradation rates caspase-inhibitor complex. In the Schliemann model, the same parameters are sensitive, plus additionally the Caspase-6 degradation rate and the inactivation rate of the TNF receptor complex I by Caspase-3. The Albeck model has only two sensitive parameters, namely the Caspase-6 Procaspase-8 complex formation rate and the Caspase-6 degradation rate. Thus, in all three models the loop gain is sensitive to reaction rates of the positive feedback loop, namely the activation of a procaspase by a caspase. A peculiar property of the Albeck model is that the caspase inhibitors do not play an important role in the loop gain. Increasing the kinetic parameter of reactions involved in mutual activation of caspases decreases the time of death, thus leads to negative sensitivity values, cf. the blue coloured entries in Table. A reduction of the strength of this loop has the opposite effect, as can be seen from the higher degradation rate of Caspase 6, an intermediate in the activation of Caspase 8 by in the models of Albeck and Schliemann, see magenta entries in Table. A similar effect on sensitivity to increasing the mutual activation is obtained by decreasing the inhibition, e.g. by increasing the breakup of caspase-inhibitor complex, Copyright 5 IFAC 958

5 shown in green in Table. The opposite effect is obtained by increasing the inhibition, either directly by making the binding probability higher (in red) or by degrading the inhibitor-caspase complex (dark red). While the former are reversible, the latter are irreversible. The only sensitive reaction not involved in the caspase activation feedback or caspase inhibition is the inhibition of the TNF receptor by, a reaction only present in the Schliemann model. This corresponds to an outer loop negative feedback. Table. Most sensitive parameters for loop gain. s and the base logarithm of the relative sensitivity for a ten-fold decrease of individual parameters. Reaction in blue increase mutual activation of caspases, while those in magenta decrease it. Breakup of caspase-inhibitor complex is in green, while formation or degradation of these complexes is in red or dark red, respectively. (a) Eißing Model. value reaction k a. Caspase-3 activation k a. Caspase-8 activation k a3.99 Caspase-3 complex formation k a7.96 Caspase-3 complex degradation k a.98 BAR Caspase-8 complex formation k a3.96 BAR Caspase-8 complex degradation k d3. Caspase-3 complex dissociation k d.8 BAR Caspase-8 complex dissociation (b) Schliemann Model. value reaction k a73. Caspase-6 degradation k a7.7 Caspase-3 complex degradation k a76.96 BAR Caspase-8 complex degradation k a79. Caspase-3 activation k a8. Caspase-8 activation k a8. Caspase-3 complex formation k a8. Caspase-3 complex dissociation k a85.78 Caspase-3 inhibits TNFR Complex k a88.98 BAR Caspase-8 complex formation k d8. Caspase-3 complex dissociation k d88.5 BAR Caspase-8 complex dissociation (c) Albeck Model. value reaction k a.99 Casp6:pCasp8 complex formation k a59. Casp6 degradation 5. REVERSAL OF APOPTOSIS During the transient phase, a pro-apoptotic signal can be countered by an anti-apoptotic one in the form of a parametric perturbation. This is particularly the case for the Eißing and Schliemann models, to a lesser extend for the Albeck model, see Figure 5. Also, the larger the perturbation, the later it can be applied. If the perturbation is too small or applied to late, cell death still occurs, but with a longer delay. In the Albeck case, the activity of the mitochondria stimulates sufficiently to cause cell death, almost independently of the loop gain. a a a 3 a 7 a a 3 d 3 d. a a 59 5 (c) Albeck model a 73 a 768 a 88 a 7 a 8 a a a a a 76 8 a 88 73a7 a 8 a a 85 a 8 79 d 8 d (d) Zoom of Figure b. d 8 d88 Figure. Sensitivity of the Loop Gain. Each parameter is scaled by. and the loop gain plotted, normalised by the nominal case. Thus, parameters not affecting the loop gain yield =. s for the forward direction are denoted a i with i the reaction number, while d stands for the reverse direction. In the following we perturb at two hours after the death stimulus the two parameters the loop gain of the Eißing and Schliemann models is the most sensitive to. For both models, the two most sensitive parameters are modified multiplicatively to increase the gain (Figure 6) or to decrease it (Figure 7). Clearly, this parameter change impacts on the time of death, delaying it in Figure 6 or pushing it forward (Figure 7). In all cases, the dropping of the loop gain below one corresponds with the decision of cell death or survival being taken. The numerical values of the time when the loop gain drops below one and when increases above (Eißing) or. (Schliemann) is shown in Table 3. Table 3. Time of death and of loop gain dropping below one. δu denotes the input strength relative to the nominal case (5 for Eißing and. for Schliemann), δp the multiplicative scaling factor applied h post trigger to the two most sensitive parameters (as in Figure 5). The loop gain drops below one at t lg while t corrresponds to the time of death, defined as in Figure 5. δu δp t lg [h] t [h] δu δp t lg [h] t [h] Thus, for the two models with asymptotically stable life steady state, the loop gain sensitivity gives a hint towards parameters important to reverse the death decision making during the transition delay phase. This will be the basis for further experiments. Copyright 5 IFAC 959

6 Scaling Factor [log].5 3 T [h] Scaling Factor [log].5.5 T [h] Scaling Factor [log] 3 6 T [h] (c) Albeck model (d) Colour bar Figure 5. Impact of parametric perturbation on the Time of death. For the Eißing and Schliemann models, both caspase-inhibitor complex dissociation rates (k d3 and k d or k d8 and k d88 ) are scaled by a common factor plotted on the y-axis. For the Albeck model, the Caspase 6-Procaspase 8 association rate (k a was decreased by the factor shown on the y-axis while simultaneously the Caspase 6 degradation rate increased by the reciprocal of that factor. This parameter scaling occurs at the time after death stimulus ( T, x-axis). Black; time points after the occurrence of apoptosis, white: recovery signals leading to no cell death within 3 hours. Here death is defines as crossing (Eißing and Albeck) or.3 (Schliemann) Time of Death [h] 5 Figure 6. Relation between loop gain and key components (apoptosis decrease). Time course simulation of key components and its inhibitor. A red vertical bar in indicates when the loop gain drops below one. Nominal case: solid line, scaling of the two most sensitive parameters (see Figure 5) h post trigger by a multiplicative perturbation of.7 (dashed),.5 (dash-dotted). 6. CONCLUSION Uncovering key regulatory loops and parameters affecting the occurrence and timing of cell death is crucial for a deeper understanding of apoptosis signalling. This paper proposes to study the gain of the positive feedback loop, not directly, but as the loop gain of two coupled subsystems. Our contribution posits the idea of shaping the Figure 7. Relation between loop gain and key components (apoptosis increase). Time course simulation of key components and its inhibitor. A red vertical bar indicates that the loop gain drops below one. Nominal case: solid line, scaling of the two most sensitive parameters (as in Figure 5) h post trigger by the multiplicative factor of. ( ) or (..). loop-gain to modulate apoptosis. This approach to closedloop control is preliminary, but as the field is evolving quickly, real-time feedback control of intracellular biochemical pathways may soon become possible in vivo. REFERENCES J. G. Albeck et al. (8). Quantitative analysis of pathways controlling extrinsic apoptosis in single cells. Molecular Cell 3., 5. T. Eißing et al. (). Bistability analyses of a caspase activation model for receptor-induced apoptosis. Journal of Biological Chemistry 79.35, D. Hanahan and R. A. Weinberg (). The hallmarks of cancer. Cell., H. Huber, E. Bullinger and M. Rehm (9). Systems biology approaches to the study of apoptosis. Essentials of Apoptosis. nd ed., A. W. Kao et al. (). A neurodegenerative disease mutation that accelerates the clearance of apoptotic cells. Proceedings of the National Academy of Sciences 8., 6. L. Neumann et al. (). Dynamics within the CD95 death-inducing signaling complex decide life and death of cells. Molecular Systems Biology, 6. M. C. Readman, M. Schliemann, D. Kalamatianos and E. Bullinger (3). A feedback control perspective on models of apoptosis signal transduction. Chaos, Solitons & Fractals 5. Special Issue Functionality and Dynamics in Biological Systems, M. Rehm, H. J. Huber, H. Dussmann and J. H. Prehn (6). Systems analysis of effector caspase activation and its control by X-linked inhibitor of apoptosis protein. EMBO Journal 5.8, M. Schliemann et al. (). Heterogeneity reduces sensitivity of cell death for TNF-stimuli. BMC Systems Biology 5.,. J. E. Shoemaker and F. J. Doyle III (8). Identifying fragilities in biochemical networks: robust performance analysis of Fas signaling-induced apoptosis. Biophysical Journal 95.6, L. Trotta, E. Bullinger and R. Sepulchre (). Global analysis of dynamical decision-making models through local computation around the hidden saddle. PLoS One 7.3, e33. Copyright 5 IFAC 96