BACKGROUND
many signals initiating programmed cell death originate from specific subcellular sites or organelles, and thus require to be forwarded across intracellular space to trigger cellular suicide.
activated death receptors localize in distinct lipid raft micro domains in the plasma membrane for efficient formation of death inducing signalling complexes  <cit> .
these sites represent spatially confined regions from which death signals may emanate, either in the form of activated initiator caspases-8/- <dig> which can directly activate effector caspase- <dig>  or in the form of bid, a pro-apoptotic cytosolic bh3-only protein of the bcl- <dig> super family which is cleaved and activated by caspases-8/ <dig>  <cit> .
similarly, bh- <dig> only proteins such as bmf and bim were shown to be associated to distinct cytoskeletal structures and are released during intrinsic apoptosis induced by various stimuli  <cit> .
the cell death signals of these apoptosis inducers can be considered to spread through the cytosol by diffusion, analogous to diffusible signalling molecules in second messenger systems, and are integrated at the mitochondria, culminating in the permeabilisation of their outer membrane by pores comprised of activated bax/bak molecules  <cit> .
indeed, we and others recently showed in hela cervical cancer cells that apoptotic mitochondrial permeabilisation during extrinsic and intrinsic apoptosis can be subject to remarkable spatial coordination, resulting in waves of mitochondrial cytochrome-c  release progressing through the cell body  <cit> .
evidence has been provided also from other experimental systems that the process of momp can occur heterogeneously throughout the cell body  <cit> .
besides spatially inhomogeneous or polarized formation of caspase-8/- <dig> activation platforms during death receptor-induced apoptosis  <cit> , kinase activities which may modulate the accessibility of caspase-8/- <dig> substrate bid appear to play a role in regulating the spatial progression of mitochondrial permeabilisation  <cit> .
mitochondrial waves were also observed during intrinsic apoptosis induced by various other stimuli such as ceramide, staurosporine or direct pharmacological bak activation  <cit> .
depending on the cell type investigated, spatial signal spread in these scenarios may at least in part depend on ca2+ signalling, kinase activities and/or also on additional, presently unknown signalling processes which also might determine the initiation site of momp.
the spatial progression of mitochondrial permeabilisation therefore appears to be a common and frequent feature in multiple signalling paradigms during apoptosis initiation.
as mitochondrial bax/bak release pores were described to be rather non-selective  <cit> , spatial momp waves can also be expected for other pro-apoptotic proteins besides cyt-c which are released from mitochondria, such as the xiap antagonists smac/diablo and htra2/omi  <cit> .
once released, these proteins are able to initiate the activation of effector caspases, the main executioners of apoptosis.
presently, it is not known whether the occurrence of spatial anisotropies in mitochondrial permeabilisation correlates with a spatially coordinated or targeted activation of effector caspases  <cit> .
previous single-cell imaging and mathematical modelling studies of apoptosis execution provided valuable insight into the temporal signalling kinetics and systems properties of the apoptosis execution network but ignored diffusion processes  <cit> .
building on this, we here therefore developed a reaction-diffusion model of the apoptosis execution network to investigate how protein diffusion impacts not only on the temporal but also on the spatial coordination of apoptotic cell death.
this also served as an in silico estimation as to whether a functional link could exist between previously reported biological findings on spatially inhomogeneous apoptosis initiation and targeted or locally restricted of apoptotic executioner caspase activities  <cit> .
RESULTS
generation of a reaction-diffusion model of apoptosis execution for hela cervical cancer cells
to investigate how spatial anisotropies during mitochondrial permeabilisation and cyt-c release reflect in the subsequent response of cytosolic effector caspase activation we performed an in silico analysis of the signalling dynamics using a spatial systems model.
this model describes the reaction network of cyt-c initiated apoptosis execution according to a previously described network topology  <cit> .
in the model, cyt-c and smac release were implemented as processes independent of postmitochondrial feed-back by executioner caspases  <cit> .
released cyt-c then triggers apoptosome formation, whose absolute concentration was limited by the total amount of available procaspase- <dig> or apaf- <dig>  <cit> .
cyt-c concentrations in this scenario were assumed not to limit apoptosome formation  <cit> .
the kinetics of cyt-c induced apoptosome formation and smac release were modelled with kinetics determined experimentally previously  <cit>  .
from these inputs the model calculated the resulting effector caspase activation profile by assuming the following biochemical processes: procaspase- <dig> bound to the apoptosome can auto-catalytically process itself to its p35/p <dig> form and can activate procaspases- <dig> and - <dig> by proteolysis  <cit> .
active caspase- <dig> can process caspase- <dig> to its p35/p <dig> form in positive feedback  <cit> .
x-linked inhibitor of apoptosis protein  was implemented as an inhibitor of caspase- <dig>  and caspases- <dig> and - <dig>  <cit> .
in reverse, xiap can be cleaved by caspase- <dig> into bir1- <dig> and bir3-ring cleavage fragments  <cit> .
xiap binding partners were assumed to be ubiquitinated and targeted for enhanced proteasomal degradation  <cit> .
effector caspase activity furthermore was implemented to impair protein synthesis and protein degradation  <cit> .
substrate cleavage by effector caspases represented the output of the model, allowing comparison of model outputs with experimental traces obtained from the cleavage of recombinant caspase substrates  <cit> .
a non-spatial reaction network on the basis of mass action kinetics for the above delineated processes was already described previously as a set of ordinary differential equations   <cit> .
while this ode model neglected diffusion processes, it was sufficient to reliably represent whole cell kinetics of apoptosis execution as experimentally recorded at minutes resolution in hela cervical and mcf- <dig> breast cancer cells  <cit> .
as fast diffusion-adsorption processes likely contribute to the subcellular spatial spread of mitochondrial permeabilisation  <cit> , we developed a spatial extension of the ode model towards a reaction-diffusion network comprised of partial differential equations .
a graphical representation of the model is shown as fig.
<dig>  all state variables and associated reactions are provided in additional file  <dig>  tables s <dig> and s <dig>  to implement diffusion in the model, we assigned diffusion coefficients d to the individual proteins in the reaction network based on their molecular masses.
the diffusion coefficients were calculated by the stokes-einstein relation in reference to the diffusion coefficient of a  <dig> kda gfp protein which was previously determined as dgfp =  <dig> μm2s- <dig> in cellular cytosolic environments  <cit> :
the mass for all proteins and protein complexes as well as their calculated diffusion coefficients are listed in additional file  <dig>  table s <dig>
concentration changes ∂cn of the individual reacting proteins and protein complexes n by diffusion in time and space were implemented obeying fick's second law for non-steady state diffusion in one dimension  <cit> .
modelling was performed in one spatial dimension to balance the abstraction of the biological scenario versus a mathematically manageable systems model .
to qualitatively estimate the error incurring from the dimension reduction, we compared diffusive signal progression in one and three spatial dimensions for a simplified diffusion scenario .
the spatiotemporal discrepancies in diffusive spread between the two scenarios were very small and suggested that one-dimensional modelling could be employed to investigate the reaction-diffusion processes during apoptosis execution.
the network was initiated by two first order exponential saturation functions , representing pseudo-reactions for cyt-c dependent apoptosome formation and smac release into the cytosol.
highest possible spatial anisotropies of these two processes due to mitochondrial permeabilisation waves were implemented by modelling that apoptosis is initiated first at one end of a cell and then travels to the other end by assigning location-dependent onset times.
with tmomp =  <dig> at the point of origin x <dig> =  <dig> and vwave being the cellular end-to-end velocity of the spatial spread of mitochondrial permeabilisation  <cit> .
once initiated, each reaction partner cn distributes in the cytosol according the reaction-diffusion equation
which represents the extension of fick's second law for non-steady state diffusion by .
this summand describes the biochemical reaction velocities of proteins and protein complexes n according to mass action kinetics .
the modelled distance l was limited to  <dig> μm by assigning flux derivatives as zero at the near and far end boundaries as in eq.
<dig> and resembles the average length of hela cervical cancer cells.
the calculations were discretized into  <dig> steps along the modelled distance.
analogous to the ode model, the pde model provided cleavage kinetics of a caspase- <dig> substrate as an output for comparison to experimentally obtained spatiotemporal data of caspase- <dig> substrate cleavage in single living cells.
mathematical modelling suggests that spatial anisotropies during apoptotic mitochondrial permeabilisation are lost upon activation of effector caspases
we initially set out to mathematically determine the influence of protein diffusion on the spatiotemporal activation profile of effector caspases.
to resemble experimentally determined slow waves of apoptotic mitochondrial permeabilisation  <cit>  , the input functions for cyt-c induced apoptosome formation and smac release were triggered with propagation velocities of  <dig>  μm/s.
for a modelled cell of  <dig> μm length the local inputs of cyt-c induced apoptosome formation and cytosolic smac hence were delayed by  <dig> min between the near and far ends .
according to previous experimental analyses  <cit> , smac release locally proceeded with slower kinetics than cyt-c induced apoptosome formation .
when analyzing the resulting spatiotemporal profiles of substrate cleavage by effector caspases, the spatial anisotropies of the model input were translated into a homogeneous response across the modelled distance .
in contrast, upon running the simulation in absence of diffusion, the spatiotemporal delays of the momp wave as expected were maintained .
corresponding results were obtained for the calculated amounts of free active effector caspase- <dig> in presence and absence of diffusion .
of note, protein diffusion caused a delay in effector caspase activity at the near end, and an earlier occurrence of caspase activity at the far end when compared to the model in absence of diffusion .
even though spatial waves of apoptotic permeabilisation of the outer mitochondrial membrane can be observed experimentally in hela cervical cancer cells, these simulations suggested that cytosolic diffusion may eliminate these spatiotemporal anisotropies during apoptosis execution.
the elimination of spatial anisotropies during apoptosis execution is highly robust to impaired protein diffusivity
protein diffusivity depends not only on particle size , but also on temperature and the viscosity of the intracellular microenvironment.
these parameters cannot be meaningfully altered in experimental settings without massively compromising cellular physiology and vitality.
theoretical systems modelling therefore is particularly appropriate to evaluate the power of molecular diffusion in eliminating spatial anisotropies during apoptotic signalling.
we employed the in silico hela cell model to investigate to which extent diffusion needs to be impaired to allow for non-homogeneous signalling to persist from mitochondrial permeabilisation until effector caspase activation.
we therefore recalculated the spatiotemporal signalling kinetics at reduced protein diffusivities and compared end-to-end delays in onset of substrate cleavage  and half maximal substrate cleavage.
spatial asynchronies in substrate cleavage could be observed upon 100-fold reduction of diffusion, while  <dig> -fold reduced diffusion coefficients were required to obtain end-to-end delays closely resembling conditions in the absence of diffusion .
a subsequent, more detailed analysis confirmed that the early onset phase would be most prone to spatial anisotropies, with the onset of substrate cleavage by effector caspases becoming asynchronous upon 25-fold impaired diffusion .
of note, no major difference in end-to-end delays could be observed for higher amounts of cleaved substrate .
reducing diffusion by up to  <dig> orders of magnitude and plotting the full profiles of substrate cleavage at the near and far ends correspondingly showed the noticeable temporal separation of substrate cleavage profiles upon approx.
100-fold reduced diffusivity .
these findings therefore demonstrate the strong influence of diffusion in eliminating spatial anisotropies within a closed cellular system.
to further explore the signalling system for processes that potentially contribute to or counteract a spatially synchronous execution of apoptosis, we investigated the role of caspase-dependent feed-backs, altered reaction kinetics and the consequences of immobilising macromolecular aggregates.
the role of caspase-dependent feed-backs in spatiotemporal apoptotic signalling
systems analysis of the mapk pathways recently suggested feed-back circuits to contribute to the rapid spatial progression of phosphoprotein signals  <cit> .
in analogy, feed-back amplification loops during apoptosis execution, such as detected between caspase- <dig> and caspase- <dig>  likewise could contribute to the swift spatial spread of death signals.
to obtain further insight into the role of caspase feed-back for a spatially homogeneous response of apoptosis execution in the hela cell systems model, we investigated scenarios in which either the caspase- <dig> feedback onto caspase- <dig>  or the capase- <dig> auto-processing were blocked.
eliminating these feedbacks resulted in reduced amounts of processed caspases, an overall delay in substrate cleavage , and comparable end-to-end anisotropies in substrate cleavage between both scenarios .
if caspase feed-backs had a role in promoting spatial anisotropies, we expected their inhibition to result in a reduction of end-to-end delays when compared to the reference model .
reductions could indeed be observed , however these were rather modest if not negligible.
increased biochemical reactivity and immobilisation of apoptosome species promote spatial anisotropies
the rate constants for the reactions included in our systems model largely were determined by quantitative biochemical experimentation using purified reactants at low total protein concentrations in aqueous reaction buffers .
reaction rates in macromolecule-rich environments such as the cytosol can substantially differ from those observed in vitro due to the volume exclusion arising from macromolecular crowding  <cit> .
volume exclusion therefore can potentially accelerate the kinetics of protein-protein interactions and enzymatic catalyses, given that the reactants are present in sufficient abundance  <cit> .
we therefore repeated the preceding calculations of end-to-end delays in substrate cleavage  in a variant of the hela cell model that employed 10-fold accelerated reaction kinetics.
while substrate cleavage at normal diffusivity still proceeded synchronously throughout the cell, end-to-end delays now became apparent upon ≥ 10-fold reduced diffusivities .
comparison of the end-to-end delays to the reference model highlighted that anisotropies in the onset of substrate cleavage  were only mildly affected, whereas higher spatial heterogeneities displayed for half-maximal substrate cleavage .
the diffusive mobility of very large proteins and protein complexes  was shown to be significantly impaired in dense cytosolic environments containing microcompartments  <cit>  and could thereby promote spatial heterogeneities.
we therefore also simulated a scenario in which the apoptosome  and all apoptosome-bound protein species were considered immobile .
in this case, anisotropy discrepancies to the reference model were most pronounced for the onset of substrate cleavage .
a combination of 10-fold increased biochemical reactivity and apoptosome immobilisation further enhanced the calculated spatial heterogeneities  and resulted in considerable anisotropies both in the onset of substrate cleavage as well as in half-maximal substrate cleavage .
as expected, in all scenarios discrepancies to the reference model were rather small at strongly impaired diffusivity  , as for this condition high spatial anisotropies already existed in the reference scenario .
taken together these data indicate that acceleration of biochemical reactions arising from volume exclusion as well as immobilisation of large macromolecular aggregates such as the apoptosome could promote spatial anisotropies during apoptosis execution.
experimental validation of model prediction: spatial anisotropies during mitochondrial permeabilisation are eliminated upon apoptosis execution in hela cervical cancer cells
we next validated experimentally in hela cervical cancer cells whether effector caspase activity sets on synchronously throughout the cytosol.
to this end, we combined mitochondrial protein redistribution measurements with a fret-based analysis of effector caspase activity in individual cells.
the fret probe consists of cyan fluorescent protein  and yellow fluorescent protein  interconnected by a short linker containing the preferred caspase- <dig> recognition site devd  <cit> .
hela cells expressing the devd fret probe were co-transfected with a plasmid coding for expression of a monomeric red fluorescent protein targeted to the mitochondrial intermembrane space .
this probe was previously shown to be released from the mitochondria into the cytosol similarly to cytochrome-c and smac during momp  <cit>  and served to identify mitochondrial permeabilisation waves .
imaging was performed by employing a previously described technique that allowed rapid sampling in absence of noticeable photobleaching or phototoxicity  <cit> .
we achieved a temporal resolution of as little as  <dig> sec when sampling cfp, fret, yfp, and rfp channels for individual cells, representing a 30-fold higher temporal resolution than previous time-lapse imaging studies.
the temporal resolution could not be further increased due to the time required for physically moving the optical components in the instrument to acquire the different channels.
following apoptosis induction with  <dig> ng/ml trail/ <dig> μg/ml chx, we observed temporal delays in the onset of ims-rp release from mitochondria into the cytosol between the near and far ends of individual cells, thus confirming that mitochondrial permeabilisation could proceed anisotropically .
in contrast, effector caspase activation always occurred simultaneously in both regions .
lipotransfection to achieve sufficiently high expression of the ims-rp was accompanied by some spontaneous cell death of hela cells.
to exclude cellular stress as a factor influencing the experimental results, we aimed to further validate our finding in a larger number of hela cells without expression of this probe.
we previously identified that in response to  <dig> μm sts or trail/chx  spatial anisotropies of mitochondrial permeabilisation can be detected in up to 70% of cells  <cit> .
following sts or trail/chx treatments, individual cells were analyzed for effector caspase activation in regions at opposite ends of the cell body.
in all cells analyzed  fret probe cleavage started simultaneously at opposite ends of individual cells .
these experimental data therefore confirm the prediction if the reference model that spatial anisotropies during the initiation of mitochondrial permeabilisation are efficiently translated into a homogeneous response of effector caspase activation throughout the entire cytosol.
discussion
our mathematical estimation of the spatial dynamics during apoptosis execution suggests that diffusion of the multiple proteins and protein complexes involved in the execution network would equalize any spatial anisotropies that were initially present during the time period required from cyt-c release to effector caspase- <dig> activation.
fast sampling of effector caspase activation confirmed experimentally that even though momp can be spatiotemporally organized, the signalling network during apoptosis execution translates this into a homogeneous response of effector caspase activation.
the mathematical reference model we devised is simplistic in that it omits the explicit consequences of macromolecular crowding and diffusion barriers: with respect to diffusion the cytosol cannot be considered to be a homogeneous aqueous environment as it is densely crowded by organelles and macromolecules.
the macro-molecularly excluded volume was estimated as 20-30%, resulting in up to 3-fold reduced diffusion coefficients for both small and large molecules in eukaryotic cells  <cit> .
as our calculations of protein diffusion built on a reference diffusion coefficient for green fluorescent protein measured in an intracellular environment  <cit> , these factors can be considered to be taken into account at least partially, while impaired diffusivity of very large protein complexes that may violate the stokes-einstein relation were investigated separately .
we found that lack of caspase feedbacks had negligible influence on spatiotemporal asynchronies during apoptosis execution, which might seem counterintuitive given that feed backs in other scenarios were shown to play a significant role in the spatial spread of protein signals  <cit> .
as absence of caspase feedbacks results in an overall slower signalling during apoptosis execution, the prolonged lag time between momp and substrate cleavage allows for diffusion to efficiently eliminate spatial anisotropies.
of note, our modelling was performed assuming one-dimensional reaction-diffusion processes as anisotropic three-dimensional scenarios currently cannot be handled mathematically by the available modelling tools.
even though for a simplified scenario of one diffusing protein largely identical spatiotemporal spreads were calculated , we would expect that for biologically more authentic cases of three-dimensional, non-symmetric adherent cells further distortions between both scenarios could become apparent.
the mathematical model also assumes strict location-dependent onset times of mitochondrial permeabilisation, whereas it can be conceived that in a living cell system the propagation process might naturally be subject to variability.
furthermore, additional signalling events may exist that affect the spatial spread of signalling in different cell types or cells lines and which are not covered in our model.
nevertheless, the one-dimensional modelling approach still was sufficiently accurate to predict the subsequent experimental finding that spatial anisotropies would be eliminated at standard conditions in hela cells.
as shown in this study by modelling impaired diffusivity, more than 100-fold lowered diffusion coefficients were required to maintain spatial anisotropies during apoptosis execution in the reference scenario.
in many signal transduction systems the activation and deactivation of the respective signalling molecules are spatially separated to establish or maintain spatial anisotropies in presence of diffusion, such as for example in the epidermal growth factor receptor/phosphotyrosine phosphatase 1b and the mitogen-activated protein kinase  signalling systems  <cit> .
when assuming that both upstream and downstream proteins during apoptosis execution are not locally retained or produced/degraded, it seems reasonable that spatial anisotropies can only be maintained upon strong impairment of diffusivity.
in addition, cytoplasmic agitation by molecular motors was recently suggested to further enhance cytosolic molecular motions and thus to promote the efficiency of spatial homogenisation of otherwise solely diffusive processes  <cit> .
while diffusion can ensure the efficient execution of cell death during canonical apoptotic signalling, several studies suggested that in certain scenarios effector caspase activity can be targeted selectively to specific subcellular regions and to limited sets of substrates.
for example, effector caspase activity targeted exclusively towards the nucleus seems to be required for the differentiation of lens cells, erythrocytes, and megakaryocytes  <cit> .
our data would suggest that in these particular situations additional mechanisms must exist that  limit and confine active caspases to target regions and/or  protect diffusing soluble caspase substrates from unwanted proteolysis.
localised accumulation or compartmentalisation of active caspases by as of yet unknown anchoring mechanisms indeed has been reported before  <cit> .
however, as literally hundreds of known protein substrates, many of them soluble, can be cleaved by caspases  <cit>  the control of local, substrate-selective caspase activity must be subject to further control mechanisms.
additional levels of control could rely on regulating caspase activities and substrate availabilities by  posttranslational modifications.
the susceptibilities of several caspase substrates were previously shown to be modulated by phosphorylation, as were the activities of caspases- <dig> and  <dig> themselves  <cit> , suggesting that kinase/phosphatase signalling could be crucial for both direct caspase regulation as well as for restricting or specifying substrate availabilities by modulating the cellular phosphoproteome.
as recently shown for the drosophila effector caspase drice, a homologue of human effector caspases- <dig> and - <dig>  ubiquitin conjugation could likewise modulate activities by impairing substrate access to the catalytic site of the enzyme  <cit> .
in the light of the rapidness of multi-protein diffusion during apoptosis execution, our study therefore also highlights the physiological significance of these control mechanisms, which so far have been largely ignored in spatially restricted, sublethal caspase activation scenarios.
CONCLUSIONS
our study highlights that diffusion of the multiple proteins constituting the apoptosis execution network is sufficient to robustly eliminate the spatial asynchronies that can be observed during the initiation of apoptosis.
it is therefore highly unlikely that anisotropic initiation of apoptosis is linked mechanistically to scenarios in which effector caspase activities were reported to be subcellularly coordinated or confined.
a homogeneous activation of effector caspases throughout the entire cell body due to diffusion might indeed be an important contributor to the efficiency of apoptotic cell clearance.

 many signals initiating programmed cell death originate from specific subcellular sites or organelles, and thus require to be forwarded across intracellular space to trigger cellular suicide. activated death receptors localize in distinct lipid raft micro domains in the plasma membrane for efficient formation of death inducing signalling complexes. these sites represent spatially confined regions from which death signals may emanate, either in the form of activated initiator caspases-8/- which can directly activate effector caspase- or in the form of bid, a pro-apoptotic cytosolic bh3-only protein of the bcl- super family which is cleaved and activated by caspases-8/. similarly, bh- only proteins such as bmf and bim were shown to be associated to distinct cytoskeletal structures and are released during intrinsic apoptosis induced by various stimuli. the cell death signals of these apoptosis inducers can be considered to spread through the cytosol by diffusion, analogous to diffusible signalling molecules in second messenger systems, and are integrated at the mitochondria, culminating in the permeabilisation of their outer membrane by pores comprised of activated bax/bak molecules. indeed, we and others recently showed in hela cervical cancer cells that apoptotic mitochondrial permeabilisation during extrinsic and intrinsic apoptosis can be subject to remarkable spatial coordination, resulting in waves of mitochondrial cytochrome-c release progressing through the cell body. evidence has been provided also from other experimental systems that the process of momp can occur heterogeneously throughout the cell body. besides spatially inhomogeneous or polarized formation of caspase-8/- activation platforms during death receptor-induced apoptosis, kinase activities which may modulate the accessibility of caspase-8/- substrate bid appear to play a role in regulating the spatial progression of mitochondrial permeabilisation. mitochondrial waves were also observed during intrinsic apoptosis induced by various other stimuli such as ceramide, staurosporine or direct pharmacological bak activation. depending on the cell type investigated, spatial signal spread in these scenarios may at least in part depend on ca2+ signalling, kinase activities and/or also on additional, presently unknown signalling processes which also might determine the initiation site of momp. the spatial progression of mitochondrial permeabilisation therefore appears to be a common and frequent feature in multiple signalling paradigms during apoptosis initiation. as mitochondrial bax/bak release pores were described to be rather non-selective, spatial momp waves can also be expected for other pro-apoptotic proteins besides cyt-c which are released from mitochondria, such as the xiap antagonists smac/diablo and htra2/omi. once released, these proteins are able to initiate the activation of effector caspases, the main executioners of apoptosis. presently, it is not known whether the occurrence of spatial anisotropies in mitochondrial permeabilisation correlates with a spatially coordinated or targeted activation of effector caspases. previous single-cell imaging and mathematical modelling studies of apoptosis execution provided valuable insight into the temporal signalling kinetics and systems properties of the apoptosis execution network but ignored diffusion processes. building on this, we here therefore developed a reaction-diffusion model of the apoptosis execution network to investigate how protein diffusion impacts not only on the temporal but also on the spatial coordination of apoptotic cell death. this also served as an in silico estimation as to whether a functional link could exist between previously reported biological findings on spatially inhomogeneous apoptosis initiation and targeted or locally restricted of apoptotic executioner caspase activities.  generation of a reaction-diffusion model of apoptosis execution for hela cervical cancer cells to investigate how spatial anisotropies during mitochondrial permeabilisation and cyt-c release reflect in the subsequent response of cytosolic effector caspase activation we performed an in silico analysis of the signalling dynamics using a spatial systems model. this model describes the reaction network of cyt-c initiated apoptosis execution according to a previously described network topology. in the model, cyt-c and smac release were implemented as processes independent of postmitochondrial feed-back by executioner caspases. released cyt-c then triggers apoptosome formation, whose absolute concentration was limited by the total amount of available procaspase- or apaf-. cyt-c concentrations in this scenario were assumed not to limit apoptosome formation. the kinetics of cyt-c induced apoptosome formation and smac release were modelled with kinetics determined experimentally previously. from these inputs the model calculated the resulting effector caspase activation profile by assuming the following biochemical processes: procaspase- bound to the apoptosome can auto-catalytically process itself to its p35/p form and can activate procaspases- and - by proteolysis. active caspase- can process caspase- to its p35/p form in positive feedback. x-linked inhibitor of apoptosis protein was implemented as an inhibitor of caspase- and caspases- and -. in reverse, xiap can be cleaved by caspase- into bir1- and bir3-ring cleavage fragments. xiap binding partners were assumed to be ubiquitinated and targeted for enhanced proteasomal degradation. effector caspase activity furthermore was implemented to impair protein synthesis and protein degradation. substrate cleavage by effector caspases represented the output of the model, allowing comparison of model outputs with experimental traces obtained from the cleavage of recombinant caspase substrates. a non-spatial reaction network on the basis of mass action kinetics for the above delineated processes was already described previously as a set of ordinary differential equations. while this ode model neglected diffusion processes, it was sufficient to reliably represent whole cell kinetics of apoptosis execution as experimentally recorded at minutes resolution in hela cervical and mcf- breast cancer cells. as fast diffusion-adsorption processes likely contribute to the subcellular spatial spread of mitochondrial permeabilisation, we developed a spatial extension of the ode model towards a reaction-diffusion network comprised of partial differential equations. a graphical representation of the model is shown as fig. all state variables and associated reactions are provided in additional file tables s and s to implement diffusion in the model, we assigned diffusion coefficients d to the individual proteins in the reaction network based on their molecular masses. the diffusion coefficients were calculated by the stokes-einstein relation in reference to the diffusion coefficient of a kda gfp protein which was previously determined as dgfp = μm2s- in cellular cytosolic environments: the mass for all proteins and protein complexes as well as their calculated diffusion coefficients are listed in additional file table s concentration changes ∂cn of the individual reacting proteins and protein complexes n by diffusion in time and space were implemented obeying fick's second law for non-steady state diffusion in one dimension. modelling was performed in one spatial dimension to balance the abstraction of the biological scenario versus a mathematically manageable systems model. to qualitatively estimate the error incurring from the dimension reduction, we compared diffusive signal progression in one and three spatial dimensions for a simplified diffusion scenario. the spatiotemporal discrepancies in diffusive spread between the two scenarios were very small and suggested that one-dimensional modelling could be employed to investigate the reaction-diffusion processes during apoptosis execution. the network was initiated by two first order exponential saturation functions, representing pseudo-reactions for cyt-c dependent apoptosome formation and smac release into the cytosol. highest possible spatial anisotropies of these two processes due to mitochondrial permeabilisation waves were implemented by modelling that apoptosis is initiated first at one end of a cell and then travels to the other end by assigning location-dependent onset times. with tmomp = at the point of origin x = and vwave being the cellular end-to-end velocity of the spatial spread of mitochondrial permeabilisation. once initiated, each reaction partner cn distributes in the cytosol according the reaction-diffusion equation which represents the extension of fick's second law for non-steady state diffusion by. this summand describes the biochemical reaction velocities of proteins and protein complexes n according to mass action kinetics. the modelled distance l was limited to μm by assigning flux derivatives as zero at the near and far end boundaries as in eq. and resembles the average length of hela cervical cancer cells. the calculations were discretized into steps along the modelled distance. analogous to the ode model, the pde model provided cleavage kinetics of a caspase- substrate as an output for comparison to experimentally obtained spatiotemporal data of caspase- substrate cleavage in single living cells. mathematical modelling suggests that spatial anisotropies during apoptotic mitochondrial permeabilisation are lost upon activation of effector caspases we initially set out to mathematically determine the influence of protein diffusion on the spatiotemporal activation profile of effector caspases. to resemble experimentally determined slow waves of apoptotic mitochondrial permeabilisation, the input functions for cyt-c induced apoptosome formation and smac release were triggered with propagation velocities of μm/s. for a modelled cell of μm length the local inputs of cyt-c induced apoptosome formation and cytosolic smac hence were delayed by min between the near and far ends. according to previous experimental analyses, smac release locally proceeded with slower kinetics than cyt-c induced apoptosome formation. when analyzing the resulting spatiotemporal profiles of substrate cleavage by effector caspases, the spatial anisotropies of the model input were translated into a homogeneous response across the modelled distance. in contrast, upon running the simulation in absence of diffusion, the spatiotemporal delays of the momp wave as expected were maintained. corresponding results were obtained for the calculated amounts of free active effector caspase- in presence and absence of diffusion. of note, protein diffusion caused a delay in effector caspase activity at the near end, and an earlier occurrence of caspase activity at the far end when compared to the model in absence of diffusion. even though spatial waves of apoptotic permeabilisation of the outer mitochondrial membrane can be observed experimentally in hela cervical cancer cells, these simulations suggested that cytosolic diffusion may eliminate these spatiotemporal anisotropies during apoptosis execution. the elimination of spatial anisotropies during apoptosis execution is highly robust to impaired protein diffusivity protein diffusivity depends not only on particle size, but also on temperature and the viscosity of the intracellular microenvironment. these parameters cannot be meaningfully altered in experimental settings without massively compromising cellular physiology and vitality. theoretical systems modelling therefore is particularly appropriate to evaluate the power of molecular diffusion in eliminating spatial anisotropies during apoptotic signalling. we employed the in silico hela cell model to investigate to which extent diffusion needs to be impaired to allow for non-homogeneous signalling to persist from mitochondrial permeabilisation until effector caspase activation. we therefore recalculated the spatiotemporal signalling kinetics at reduced protein diffusivities and compared end-to-end delays in onset of substrate cleavage and half maximal substrate cleavage. spatial asynchronies in substrate cleavage could be observed upon 100-fold reduction of diffusion, while -fold reduced diffusion coefficients were required to obtain end-to-end delays closely resembling conditions in the absence of diffusion. a subsequent, more detailed analysis confirmed that the early onset phase would be most prone to spatial anisotropies, with the onset of substrate cleavage by effector caspases becoming asynchronous upon 25-fold impaired diffusion. of note, no major difference in end-to-end delays could be observed for higher amounts of cleaved substrate. reducing diffusion by up to orders of magnitude and plotting the full profiles of substrate cleavage at the near and far ends correspondingly showed the noticeable temporal separation of substrate cleavage profiles upon approx. 100-fold reduced diffusivity. these findings therefore demonstrate the strong influence of diffusion in eliminating spatial anisotropies within a closed cellular system. to further explore the signalling system for processes that potentially contribute to or counteract a spatially synchronous execution of apoptosis, we investigated the role of caspase-dependent feed-backs, altered reaction kinetics and the consequences of immobilising macromolecular aggregates. the role of caspase-dependent feed-backs in spatiotemporal apoptotic signalling systems analysis of the mapk pathways recently suggested feed-back circuits to contribute to the rapid spatial progression of phosphoprotein signals. in analogy, feed-back amplification loops during apoptosis execution, such as detected between caspase- and caspase- likewise could contribute to the swift spatial spread of death signals. to obtain further insight into the role of caspase feed-back for a spatially homogeneous response of apoptosis execution in the hela cell systems model, we investigated scenarios in which either the caspase- feedback onto caspase- or the capase- auto-processing were blocked. eliminating these feedbacks resulted in reduced amounts of processed caspases, an overall delay in substrate cleavage, and comparable end-to-end anisotropies in substrate cleavage between both scenarios. if caspase feed-backs had a role in promoting spatial anisotropies, we expected their inhibition to result in a reduction of end-to-end delays when compared to the reference model. reductions could indeed be observed, however these were rather modest if not negligible. increased biochemical reactivity and immobilisation of apoptosome species promote spatial anisotropies the rate constants for the reactions included in our systems model largely were determined by quantitative biochemical experimentation using purified reactants at low total protein concentrations in aqueous reaction buffers. reaction rates in macromolecule-rich environments such as the cytosol can substantially differ from those observed in vitro due to the volume exclusion arising from macromolecular crowding. volume exclusion therefore can potentially accelerate the kinetics of protein-protein interactions and enzymatic catalyses, given that the reactants are present in sufficient abundance. we therefore repeated the preceding calculations of end-to-end delays in substrate cleavage in a variant of the hela cell model that employed 10-fold accelerated reaction kinetics. while substrate cleavage at normal diffusivity still proceeded synchronously throughout the cell, end-to-end delays now became apparent upon ≥ 10-fold reduced diffusivities. comparison of the end-to-end delays to the reference model highlighted that anisotropies in the onset of substrate cleavage were only mildly affected, whereas higher spatial heterogeneities displayed for half-maximal substrate cleavage. the diffusive mobility of very large proteins and protein complexes was shown to be significantly impaired in dense cytosolic environments containing microcompartments and could thereby promote spatial heterogeneities. we therefore also simulated a scenario in which the apoptosome and all apoptosome-bound protein species were considered immobile. in this case, anisotropy discrepancies to the reference model were most pronounced for the onset of substrate cleavage. a combination of 10-fold increased biochemical reactivity and apoptosome immobilisation further enhanced the calculated spatial heterogeneities and resulted in considerable anisotropies both in the onset of substrate cleavage as well as in half-maximal substrate cleavage. as expected, in all scenarios discrepancies to the reference model were rather small at strongly impaired diffusivity, as for this condition high spatial anisotropies already existed in the reference scenario. taken together these data indicate that acceleration of biochemical reactions arising from volume exclusion as well as immobilisation of large macromolecular aggregates such as the apoptosome could promote spatial anisotropies during apoptosis execution. experimental validation of model prediction: spatial anisotropies during mitochondrial permeabilisation are eliminated upon apoptosis execution in hela cervical cancer cells we next validated experimentally in hela cervical cancer cells whether effector caspase activity sets on synchronously throughout the cytosol. to this end, we combined mitochondrial protein redistribution measurements with a fret-based analysis of effector caspase activity in individual cells. the fret probe consists of cyan fluorescent protein and yellow fluorescent protein interconnected by a short linker containing the preferred caspase- recognition site devd. hela cells expressing the devd fret probe were co-transfected with a plasmid coding for expression of a monomeric red fluorescent protein targeted to the mitochondrial intermembrane space. this probe was previously shown to be released from the mitochondria into the cytosol similarly to cytochrome-c and smac during momp and served to identify mitochondrial permeabilisation waves. imaging was performed by employing a previously described technique that allowed rapid sampling in absence of noticeable photobleaching or phototoxicity. we achieved a temporal resolution of as little as sec when sampling cfp, fret, yfp, and rfp channels for individual cells, representing a 30-fold higher temporal resolution than previous time-lapse imaging studies. the temporal resolution could not be further increased due to the time required for physically moving the optical components in the instrument to acquire the different channels. following apoptosis induction with ng/ml trail/ μg/ml chx, we observed temporal delays in the onset of ims-rp release from mitochondria into the cytosol between the near and far ends of individual cells, thus confirming that mitochondrial permeabilisation could proceed anisotropically. in contrast, effector caspase activation always occurred simultaneously in both regions. lipotransfection to achieve sufficiently high expression of the ims-rp was accompanied by some spontaneous cell death of hela cells. to exclude cellular stress as a factor influencing the experimental results, we aimed to further validate our finding in a larger number of hela cells without expression of this probe. we previously identified that in response to μm sts or trail/chx spatial anisotropies of mitochondrial permeabilisation can be detected in up to 70% of cells. following sts or trail/chx treatments, individual cells were analyzed for effector caspase activation in regions at opposite ends of the cell body. in all cells analyzed fret probe cleavage started simultaneously at opposite ends of individual cells. these experimental data therefore confirm the prediction if the reference model that spatial anisotropies during the initiation of mitochondrial permeabilisation are efficiently translated into a homogeneous response of effector caspase activation throughout the entire cytosol. discussion our mathematical estimation of the spatial dynamics during apoptosis execution suggests that diffusion of the multiple proteins and protein complexes involved in the execution network would equalize any spatial anisotropies that were initially present during the time period required from cyt-c release to effector caspase- activation. fast sampling of effector caspase activation confirmed experimentally that even though momp can be spatiotemporally organized, the signalling network during apoptosis execution translates this into a homogeneous response of effector caspase activation. the mathematical reference model we devised is simplistic in that it omits the explicit consequences of macromolecular crowding and diffusion barriers: with respect to diffusion the cytosol cannot be considered to be a homogeneous aqueous environment as it is densely crowded by organelles and macromolecules. the macro-molecularly excluded volume was estimated as 20-30%, resulting in up to 3-fold reduced diffusion coefficients for both small and large molecules in eukaryotic cells. as our calculations of protein diffusion built on a reference diffusion coefficient for green fluorescent protein measured in an intracellular environment, these factors can be considered to be taken into account at least partially, while impaired diffusivity of very large protein complexes that may violate the stokes-einstein relation were investigated separately. we found that lack of caspase feedbacks had negligible influence on spatiotemporal asynchronies during apoptosis execution, which might seem counterintuitive given that feed backs in other scenarios were shown to play a significant role in the spatial spread of protein signals. as absence of caspase feedbacks results in an overall slower signalling during apoptosis execution, the prolonged lag time between momp and substrate cleavage allows for diffusion to efficiently eliminate spatial anisotropies. of note, our modelling was performed assuming one-dimensional reaction-diffusion processes as anisotropic three-dimensional scenarios currently cannot be handled mathematically by the available modelling tools. even though for a simplified scenario of one diffusing protein largely identical spatiotemporal spreads were calculated, we would expect that for biologically more authentic cases of three-dimensional, non-symmetric adherent cells further distortions between both scenarios could become apparent. the mathematical model also assumes strict location-dependent onset times of mitochondrial permeabilisation, whereas it can be conceived that in a living cell system the propagation process might naturally be subject to variability. furthermore, additional signalling events may exist that affect the spatial spread of signalling in different cell types or cells lines and which are not covered in our model. nevertheless, the one-dimensional modelling approach still was sufficiently accurate to predict the subsequent experimental finding that spatial anisotropies would be eliminated at standard conditions in hela cells. as shown in this study by modelling impaired diffusivity, more than 100-fold lowered diffusion coefficients were required to maintain spatial anisotropies during apoptosis execution in the reference scenario. in many signal transduction systems the activation and deactivation of the respective signalling molecules are spatially separated to establish or maintain spatial anisotropies in presence of diffusion, such as for example in the epidermal growth factor receptor/phosphotyrosine phosphatase 1b and the mitogen-activated protein kinase signalling systems. when assuming that both upstream and downstream proteins during apoptosis execution are not locally retained or produced/degraded, it seems reasonable that spatial anisotropies can only be maintained upon strong impairment of diffusivity. in addition, cytoplasmic agitation by molecular motors was recently suggested to further enhance cytosolic molecular motions and thus to promote the efficiency of spatial homogenisation of otherwise solely diffusive processes. while diffusion can ensure the efficient execution of cell death during canonical apoptotic signalling, several studies suggested that in certain scenarios effector caspase activity can be targeted selectively to specific subcellular regions and to limited sets of substrates. for example, effector caspase activity targeted exclusively towards the nucleus seems to be required for the differentiation of lens cells, erythrocytes, and megakaryocytes. our data would suggest that in these particular situations additional mechanisms must exist that limit and confine active caspases to target regions and/or protect diffusing soluble caspase substrates from unwanted proteolysis. localised accumulation or compartmentalisation of active caspases by as of yet unknown anchoring mechanisms indeed has been reported before. however, as literally hundreds of known protein substrates, many of them soluble, can be cleaved by caspases the control of local, substrate-selective caspase activity must be subject to further control mechanisms. additional levels of control could rely on regulating caspase activities and substrate availabilities by posttranslational modifications. the susceptibilities of several caspase substrates were previously shown to be modulated by phosphorylation, as were the activities of caspases- and themselves, suggesting that kinase/phosphatase signalling could be crucial for both direct caspase regulation as well as for restricting or specifying substrate availabilities by modulating the cellular phosphoproteome. as recently shown for the drosophila effector caspase drice, a homologue of human effector caspases- and - ubiquitin conjugation could likewise modulate activities by impairing substrate access to the catalytic site of the enzyme. in the light of the rapidness of multi-protein diffusion during apoptosis execution, our study therefore also highlights the physiological significance of these control mechanisms, which so far have been largely ignored in spatially restricted, sublethal caspase activation scenarios.  our study highlights that diffusion of the multiple proteins constituting the apoptosis execution network is sufficient to robustly eliminate the spatial asynchronies that can be observed during the initiation of apoptosis. it is therefore highly unlikely that anisotropic initiation of apoptosis is linked mechanistically to scenarios in which effector caspase activities were reported to be subcellularly coordinated or confined. a homogeneous activation of effector caspases throughout the entire cell body due to diffusion might indeed be an important contributor to the efficiency of apoptotic cell clearance. 
reaction-diffusion modelling suggested that even if strong spatial anisotropies existed during mitochondrial cytochrome c release, these would be eliminated by free diffusion of the cytosolic proteins that instantiate the apoptosis execution network. experimentally, rapid sampling of mitochondrial permeabilisation and effector caspase activity in individual hela cervical cancer cells confirmed predictions of the reaction-diffusion model and demonstrated that the signalling network of apoptosis execution could efficiently translate spatial anisotropies in mitochondrial permeabilisation into a homogeneous effector caspase response throughout the cytosol. multi-protein diffusion efficiently contributes to eliminating spatial asynchronies which are present during the initiation of apoptosis execution and thereby ensures homogeneous apoptosis execution throughout the entire cell body. mitochondrial outer membrane permeabilisation and activation of effector caspases are key processes during apoptosis signalling. further systems modelling suggested that a more than -fold impaired diffusivity would be required to maintain spatial anisotropies as observed during mitochondrial permeabilisation until the time effector caspases become activated. to investigate the consequences of these spatial anisotropies in mitochondrial permeabilisation on subsequent effector caspase activation, we devised a mathematical reaction-diffusion model building on a set of partial differential equations. for previously reported biological scenarios in which effector caspase activity was shown to be targeted selectively to specific subcellular regions additional mechanisms must exist that limit or spatially coordinate caspase activation and/or protect diffusing soluble caspase substrates from unwanted proteolysis. momp can be subject to spatial coordination in human cancer cells, resulting in intracellular waves of cytochrome-c release.
 apoptosis is an essential cell death process throughout the entire life span of all metazoans and its deregulation in humans has been implicated in many proliferative and degenerative diseases. mitochondrial outer membrane permeabilisation and activation of effector caspases are key processes during apoptosis signalling. momp can be subject to spatial coordination in human cancer cells, resulting in intracellular waves of cytochrome-c release. to investigate the consequences of these spatial anisotropies in mitochondrial permeabilisation on subsequent effector caspase activation, we devised a mathematical reaction-diffusion model building on a set of partial differential equations.  reaction-diffusion modelling suggested that even if strong spatial anisotropies existed during mitochondrial cytochrome c release, these would be eliminated by free diffusion of the cytosolic proteins that instantiate the apoptosis execution network. experimentally, rapid sampling of mitochondrial permeabilisation and effector caspase activity in individual hela cervical cancer cells confirmed predictions of the reaction-diffusion model and demonstrated that the signalling network of apoptosis execution could efficiently translate spatial anisotropies in mitochondrial permeabilisation into a homogeneous effector caspase response throughout the cytosol. further systems modelling suggested that a more than -fold impaired diffusivity would be required to maintain spatial anisotropies as observed during mitochondrial permeabilisation until the time effector caspases become activated.  multi-protein diffusion efficiently contributes to eliminating spatial asynchronies which are present during the initiation of apoptosis execution and thereby ensures homogeneous apoptosis execution throughout the entire cell body. for previously reported biological scenarios in which effector caspase activity was shown to be targeted selectively to specific subcellular regions additional mechanisms must exist that limit or spatially coordinate caspase activation and/or protect diffusing soluble caspase substrates from unwanted proteolysis. 