Computational Modeling of Subcellular Transport and Signaling

Abstract

Numerous signaling processes in the cell are controlled in microdomains that are defined by cellular structures ranging from nm to µm in size. Recent improvements in microscopy enable the resolution and reconstruction of these micro domains, while new computational methods provide the means to elucidate their functional roles. Collectively these tools allow for a biophysical understanding of the cellular environment and its pathological progression in disease. Here we review recent advancements in microscopy, and subcellular modeling based on reconstructed geometries, with a special focus on signaling microdomains that are important for the excitation contraction coupling in cardiac myocytes.

Introduction

Recent advancements in electron microscopy (EM) and light microscopy have made it possible to resolve cellular micro-anatomic structures in great detail. 3D computational models leverage new structural data to examine their role in cell signaling, and their involvement in disease. While structural data are vital to computational modeling, the localization of key proteins as well as the kinetics of signaling reactions and second messenger fluxes are equally important. An example of a signaling microdomain in cardiac cells is the Ca²⁺ release unit (CRU) that drives excitation contraction coupling (ECC). At each CRU, electrical depolarization of the cell membrane during the cardiac action potential, drives a transient intracellular Ca²⁺ release event, known as a spark [1]. The coordinated and integrated release of Ca²⁺ from several thousand CRUs gives rise to the whole cell Ca²⁺ transient that triggers myofilament activation and contraction; computational models have been crucial to uncovering the quantitative determinants of this process [2].

Continued improvements in 3D microscopy and computational models are enabling a renaissance in the analysis of intracellular signaling events in near molecular detail [3]. Further integration of high-resolution bio-imaging and advanced modeling techniques will shepherd a fundamental understanding of cellular signaling enabled by the collective behavior of myriad molecular players. Here we review recent methods for subcellular modeling based on reconstructed geometries, with a special focus on signaling microdomains that are important for the ECC in cardiac myocytes.

High-Resolution Microscopy Techniques for Structural Imaging

Recent improvements in electron and light microscopy have increased the availability of structural data involved in sub cellular signaling. [4–6]. EM tomography is well-suited for identifying 3D nano-structures such as organelles and membrane systems [4]. The pixel resolution can approach ≃ 1 nm but the total sample size is limited to a few µm in the xy-plane and some 100 nm in the z-direction [4]. Different automated acquisition techniques can be applied to increase the volume, but at the cost of reduced resolution in one or more of the principal axes [7, 8]. To identify position of single channels, EM can be combined with immunolabeling techniques, but the staining percentage is low and labeling is limited to the cell surface [9]. A new labeling approach, mini Singlet Oxygen Generator, expresses genetically-encoded fluorescent labels in live cells and may expand the range of resolvable sub cellular structures [10].

Conventional confocal light microscopy (LM) can also be used to extract 3D features, but the resolution (≃100nm) is not as good as in EM tomography [6]. With super resolution LM novel registration techniques can be used to bypass the diffraction limit of light, making it possible to identify features down to 20-nm resolution [5]. By correlating light and electron microscopy, co-labeling from LM can be used to complement the structural data from EM tomography and augment subcellular structural models with protein localization data [11].

3D computational geometries from microscopy data

Computational models that integrate sub cellular structural data often rely on meshes that denote surface topology or cellular volumes. In general, the mesh construction procedure begins with segmentation of individual image frames into regions of interest. Segmentation is either done manually, or by using automated or semi automated procedures [12], or even crowd-sourcing [13]. Automated procedures are attractive as they facilitates mass processing of large microscopy data sets. However these methods are highly dependent on contrast and resolution, and may ultimately require some level of manual segmentation [8].

The segmented 3D model is suitable for visualization and for measuring quantities such as distances, areas and volumes, however the quality of the mesh is generally too poor for numerical simulations. This motivates the need for mesh refinement tools that render segmented 3D models into high quality meshes [14–17]. The surface mesh improvement library, GAMer, has enabled sub cellular simulations of Ca²⁺ signaling in cardiomyocytes [18–21] [22]. GAMer is a light weight C-library providing smoothing and coarsening algorithms suitable for improving the quality of surface meshes. The library is wrapped in Python, making it possible to be used as a plug-in in other mesh-manipulation visualization tools such as Blender (www.blender.org). Such visualization tools allow annotation of functionally-important regions in the surface mesh, e.g the localization of subcellular fluxes important for the computational models. MCell, a particle-based simulation tool, which also use Blender as a GUI, provides a number of surface mesh improvement methods of its own. In a recent study where the precise volume and structure of the extracellular volume were important, a method for correcting overlapping and incorrect segmented contours was developed [23].

Instead of generating a computational mesh, necessary for FEM or particle-based simulations, segmented images can be used directly in software packages such as VirtualCell [24]. Here subcellular regions reconstructed from imaging data can be integrated via a GUI with complex reaction schemes coupled to compartmental models. An advantage of the compartmentalized models supported by VirtualCell is the ease of building complex, but well constrained models that are informed by experimental data [25].

An Example from Cardiac Myocyte Excitation-Contraction Coupling

With early 2D EM images it was possible to estimate 3D structural properties such as the size and distribution of CRUs, which guided the construction of numerous of different subcellular models of intracellular Ca²⁺ cycling [2, 26]. These 3D metrics were estimated by extrapolating geometric data from serial 2D images. By assuming a regular, e.g. circular, geometry the extrapolation often overestimated the size of CRUs and the distances between them [27]. Similarly, basic assumptions of RyR packing density within each CRU led to overestimates of the number of RyR per dyad [26]. With the use of 3D EM tomography, more refined structural information of the CRU became available, revising our understanding of the CRU size and distribution [27]. The more detailed 3D measurements suggested that CRU are smaller and much more densely distributed than previously assumed, which was further confirmed by super resolution LM [28]. These new geometric observations were integrated into computational models leading to a refinement of our understanding of a single release event [29, 30]. However, these studies used a reconstructed 3D geometry and by reducing the CRU to a single compartment connected to a lumped cytosolic compartment, important structural details were lost from the analyses.

Using publicly available EM tomograms from ventricular cardiomyocytes Hake et al [18] reconstructed a 3D model of a single CRU unit (see Figure 1A). The EM tomogram was a few µm in size in the xy-plane but only 0.43 µm thick, and contained a single intact CRU with detailed structures of neighboring Sarcoplasmic reticulumn (SR), mitochondria and transverse tubules (t-tubule) (see Figure 1B). By explicitly modeling a Ca²⁺ sensitive-dye inside the SR, we established a correlation between local Ca²⁺ movement inside the SR with functional measurements [31]. The model predicted near total junctional Ca²⁺ depletion after the spark while keeping a regional Ca²⁺ reserve, reconciling previous model predictions with experimental measurements [31, 32].

Figure 1. Modeling Ca²⁺ sparks in a 3D reconstruction of a Ca²⁺ release unit (CRU).

A. 3D EM tomography data was used to segment a single Ca²⁺ release unit. Features like transverse tubule (TT), Mitochondria (Mit), and sarcoplasmic reticulumn (SR) where identified. B. The annotated computational mesh included a complex network of SR, two distinct Mit, and two distinct TTs. C. Together with junctional SR on of the TTs formed the CRU. Ca²⁺ was released from SR at the jSR boundary (red). D. Finite element simulations of a Ca²⁺ spark revealed huge spatial gradients within the CRU and its immediate vicinity. Adapted from [18].

A single t-tubule hosts several CRUs and facilitates the spread of excitation throughout the myocyte. Using reconstructed geometries of t-tubules based on LM data from rat [21] and rabbit [19], effects of geometrical variations on single t-tubules were investigated. In smaller mammals like mice and rats, t-tubules are branched and excitation follows a more complex pattern, which contrasts with the more linear t-tubules in higher mammals such as rabbits (see Figure 2A and B). By embedding a model of reconstructed t-tubules geometry into a coarser model of the whole cell Yu et al [20] constructed a multilevel model of the early stage of the ECC. Here, the coarser model provided boundary conditions to simulations on the reconstructed geometry.

Figure 2. The geometry of transverse tubule (t-tubules) is species dependent.

A. The computational geometry of the t-tubules from a rat is branched and B. linear in rabbit ventricular myocyte. From [21] and [19] respectively.

Spatially detailed subcellular ECC models using qualitative structural representation have been a useful complement to simulations with reconstructed geometries. Hatano et al [33] for instance combined a reaction-diffusion model of ECC and energy production in the mitochondria, with a mechanical model for contraction. Similarly, a CRU geometry from EM tomography was used in a recent model of a Ca²⁺ spark [34] to suggest a novel mechanism for spark termination, where the geometry of the CRU played a crucial role [35]. Using RyR distribution data from LM, Soeller et al [36] demonstrated that RyR puncta are distributed in a staggered fashion along the z-disks, which together with diffusion simulations could explain the anisotropic diffusion measured experimentally.

Combining discrete atomistic models with continuous models

The cell cytosol is crowded with proteins and other species that influence substrate diffusion through electrostatic, van der Waals and hydrodynamic interactions [37]. As a first step towards accounting for these influences in a model of ECC, Tanskanen et al [38] used a particle-based Monte Carlo method with nanometer resolution of the CRU. A simple box geometry was used to represent the geometry, but cryo-EM reconstructions of the channel proteins where included as well as the electrostatic potential (ESP), owing to the charged sarcolemmal lipid bilayer. It was shown that the protein structures and the ESP both impacted the effectiveness of the ECC. Similarly it was shown that densely-packed myofibril lattice with high resolution details of the filament proteins are important for accurate predictions of subcellular diffusion [39, 40]. By applying homogenization methods to a generated 2D geometry of the myofilaments experimentally-observed anisotropic diffusion coefficient could be explained [39]. Using a reconstructed 3D geometry of the thin filament from cryo-EM(see Figure 3), the estimated diffusion coefficient was recently refined [40]. Progress toward accounting for the ESP surrounding a protein was reported in [41], where electrostatic interactions were found to accelerate Ca²⁺ association rates to Troponin C and SR Ca²⁺-ATPase (SERCA) by orders of magnitude. The rate was, however, reduced upon including the proteins in its native environment, thin filament and lipid-bilayer, respectively.

Figure 3. Homogenization of myofilament lattice improved the estimate of the diffusion coefficient in cytosol.

An reconstruction with atomistic-resolution of the thin filaments in a myofibril. From [40].

Proteins are not only passive bystanders in subcellular modeling they also serve specific functional roles through conformational changes. Silva et al [42] were among the first to incorporate the structural changes in the potassium pump that were involved in the gated K⁺ current. By Monte Carlo sampling of the movement of the voltage gate helix, they quantified the free energy barriers for transitions between the proteins functional states. The barriers were then used to determine the rates in a Markov model of the ion channel. An alternative to this approach was recently published by the same lab [43] in which new governing equations were presented, leading to better predictions of channel openings. Extensive studies linking molecular dynamics (MD) simulations of the KcsA K⁺ channel to its selectivity for K⁺ relative to other endogenous cations were reviewed by Roux et al [44] and provide compelling evidence that ionic conduction can be predicted from first principles via MD. Several approaches have incorporated finite-volume, diffusing particles in the simulation of proteins through Brownian dynamics (BD) methods. Recent examples include BD predictions of diffusion-limited Ca²⁺ binding to TnC [45, 46] and SERCA [47], as well as current-voltage relationships for K⁺ diffusion through the inwardly-rectifying potassium channel [48]. Given the computational expense of explicit particle simulations, efforts to combine continuum and discrete simulations could potentially offer the best of both worlds, see for example Bauler et al [49].

Conclusion

The convergence of new microscopy methodologies and multi-scale modeling techniques is offering the opportunity to help elucidate the functional roles of cellular microdomains in numerous important signaling processes. The ability to directly image near atomistic resolution protein structural data has the potential to unwind the molecular basis of biological function and disease. Equally important are the modeling tools that can bridge a breadth of imaging techniques examining disparate time and length scales. In recent years, exciting progress has been made on both fronts, as well as techniques to couple the tools.

Highlights.

• Electron and light microscopy yield 3D reconstructions of signaling microdomains

• Imaging techniques provide further high-resolution structural data suitable for 3D modeling

• Advancements in computational geometry enable generation of 3D meshes suitable for numerical solutions

• Computational models with molecular details provide integral insight into cellular signaling