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The goal of the present study is to integrate different datasets in cell biology to derive additional quantitative information about a gene or protein of interest within a single cell using computational simulations. We propose a novel prototype cell simulator as a quantitative tool to integrate datasets including dynamic information about transcript and protein levels and the spatial information on protein trafficking in a complex cellular geometry. In order to represent the stochastic nature of transcription and gene expression, our cell simulator uses event-based stochastic simulations to capture transcription, translation, and dynamic trafficking events. In a reconstructed cellular geometry, a realistic microtubule structure is generated with a novel growth algorithm for simulating vesicular transport and trafficking events. In a case study, we investigate the change in quantitative expression levels of a water channel-aquaporin 4-in a single astrocyte cell, upon pharmacological treatment. Gillespie based discrete time approximation method results in stochastic fluctuation of mRNA and protein levels. In addition, we compute the dynamic trafficking of aquaporin-4 on microtubules in this reconstructed astrocyte. Computational predictions are validated with experimental data. The demonstrated cell simulator facilitates the analysis and prediction of protein expression dynamics.

Advancements in optical imaging, microscopy, and quantitative techniques in molecular biology allow the measurement of protein expression levels, localization, and dynamic trafficking events in a single cell or a population of cells. The entire “life-story” of a protein including the transcriptional activation of its gene, mRNA generation and processing, translation, transport of the mature protein to its subcellular destination and its eventual degradation dynamics can be measured with different molecular biology techniques. For example, the temporal evolution of protein expression after exposure to a pharmacological agent is commonly detected by techniques like western blotting. Quantitative methods like proteomics profiling or pulse-chase radioisotope labeling can measure the kinetic rates of transcription, translation and degradation. The dynamics of protein trafficking and transport are captured by using immunofluorescence labeling techniques or through the use of fusion proteins, where the target protein is tagged with a fluorescent molecule for visualization [

With fluorescent tags like quantum dots [

For the prototype cell simulator, we use stochastic simulations transcription and translation [

By incorporating kinetic data of protein synthesis and transport rates into the cell simulator, we can (1) predict the spatiotemporal distribution of a protein in response to a pharmacological stimulus; (2) quantify how a drug can alter the expression levels of a protein to better understand the mechanism behind its dynamic response; and (3) discover hidden kinetic rates that are difficult to measure experimentally using inversion methods.

In this study, we use the cell simulator to capture the transcriptional and translational mechanisms behind dynamic changes of aquaporin 4 (AQP4) expression in a single astrocyte. The aquaporin family proteins play a central role in homeostasis. Among the discovered 13 aquaporin families [

The mathematical modeling of the life-cycle of AQP4 proteins from gene activation to membrane expression is composed of sequential kinetic reaction processes for gene transcription, translation, and the transport of AQP4 proteins in vesicles towards the astrocytic endfeet. Event-based stochastic simulation mimics the random fluctuations of gene activation in a single cell. The kinetic rate constants are estimated from deterministic representation. Alternatively, kinetic rates can be determined by inversion of stochastic differential equation models [

Powerful cellular simulators exist that can describe dynamic cellular activities with deterministic or stochastic mathematical formulations. For example,

One of the advantages of modeling cellular events with a single cell simulator as opposed to using partial differential equations that describe an entire cell population is the ability to portray the trafficking and polarized protein distribution in the cell in a way that is comparable to microscopic images. Protein trafficking through vesicular transport on the cytoskeleton is a key event that brings it to its functional destination. Birbaumer and Schweitzer have simulated the dynamics of vesicle fusion and transport as stochastic transitions with the Langevin equation [

Overview of cell simulators.

Author, year | Cell simulator | Simulation event | Mathematical model | Transport on cytoskeleton | Natural cell structure | 3d Visualization | ||
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Diffusion | Reaction | Deterministic | Stochastic | |||||

Kim, 2013 | This work | ■ | ■ | ■ | ■ | ■ | ■ | ■ |

Loew, 2001 [ |
■ | ■ | ■ | ■ | ■ | ■ | ||

Tomita, 1999 [ |
■ | ■ | ■ | |||||

Hattne, 2005 [ |
■ | ■ | ■ | ■ | ■ | |||

Raymond, 2003 [ |
■ | ■ | ||||||

Ander, 2004 [ |
■ | ■ | ■ | ■ | ||||

Stiles, 1996 [ |
■ | ■ | ■ | ■ | ■ | |||

Plimpton, 2003 [ |
■ | ■ | ■ | ■ | ■ | |||

Andrews, 2004 [ |
■ | ■ | ■ | ■ | ||||

Boulianne, 2008 [ |
■ | ■ | ||||||

Le Novère, 2001 [ |
■ | ■ |

For simulating AQP4 upregulation and its endfeet polarization, our model needs to compute gene transcription, translation and protein transport in a spatially distributed system at the single cell level. For proteins that are expressed in a specific cellular location in order to carry out their physiological functions, the trafficking and polarized expression of these proteins after translation becomes very important in understanding the temporal dynamics of cellular response to a stimuli. In our particular case study, the directed transport of mature AQP4 on the microtubules toward the endfoot membrane requires an accurate mathematical representation of the packaging of AQP4 into vesicles, association with motor proteins, and realistic transport kinetics along the cytoskeleton. A major portion of protein trafficking events occurs through active transport by motor proteins along the microtubules. Taking advantage of the discrete population-based stochastic algorithm, our simulator will be able to capture the transport and reaction events of molecules in a single cell, and track the localization of each molecule at any moment in time.

The rationale of our simulator is to integrate experimental observations of cell morphology (confocal microscopy), transcription factor activation (immunofluorescence), and induced target protein expression (western blot) using a cell simulator based on stochastic simulations, and predict future behavior of the cell based on existing experimental data.

Realistic cell geometry was reconstructed from confocal microscopic images of a single astrocyte including detailed intracellular organelles such as nucleus, endoplasmic reticular, microtubules,

Cultured primary astrocytes were loaded with calcein-AM and a single stellate astrocyte was scanned with confocal microscopy using a BioRad microscope (Biorad, Hercules, CA, USA) with a Z-plane resolution of 0.5 microns. The model was then reconstructed from the stack of confocal images using an image reconstruction software, MIMICS (Materialize, Leuven, Belgium), as shown in

Reconstruction of a single astrocyte cell in 3d. A series of confocal microscopic images (

To complete the cell model, intracellular compartments and cell organelles including the nucleus, endoplasmic reticula (ER), Golgi apparatus (GA), and microtubules (MTs) were added. These cell compartments were integrated with the volumetric mesh of the astrocyte body. The nucleus surrounded by the ER was placed with mesh tools offered by Mimics as in

Implementation of nucleus (

For the growth of the microtubule cytoskeleton, we developed the smooth and shortest path finding algorithm, which aims at finding the smoothest and the shortest path in the presence of obstacles. This algorithm begins with a random directional growth of MT segment from the MTOC to the neighboring mesh. Then, it uses the directional information from the target vector,

The pseudocode for implementing Smooth and Shortest path finding algorithm to model the microtubular cytoskeleton is described as follows:

Step 1. Select a starting mesh of MT growth (the MTOC) and a target mesh located at the endfoot.

Step 2. Generate the first growth step with a random direction, with the length of the segment equal to the distance to the connecting mesh.

Step 3. Grow a segment to the neighboring mesh adjusting its direction and length with the information of tangent vector and target vector (direction to the final destination).

Step 4. Repeat step 3 until it encounters the boundary of the endfoot process.

Step 5. If the growth does not encounter the boundary of an endfoot process, the growth will be terminated and a new MT growth will start from step 1. Otherwise, the MT segment grows into the process which contains a target mesh selected in step 1.

Step 6. Grow a segment to the neighboring mesh adjusting its direction and length with the information of tangent vector, target vector, and distance to the normal surface.

Step 7. Repeat step 6 until it comes into contact with the endfoot boundary.

Step 8.Upon reaching a member of the mesh at the endfoot boundary, the smooth and shortest path finding algorithm will stop and finalize the trajectory to the target mesh selected in step 1.

Step 9. The defined end criterion for the total number of MTs will terminate the algorithm.

At each growth step, the position can be recorded as the mesh id number or as the absolute position. Each step of growth will have a slight directional and longitudinal variation to avoid running into an obstacle or the boundary of the cell. The combined consideration for path smoothness using the tangent vector and for shorter path using the distance to the closest surface from the current point will gradually modify the direction of growth.

The synthesis of AQP4 water channels involves the transcription and translation mechanisms. In our mathematical model, the transcription and translation of AQP4 under normal conditions (steady state) will be established first. In addition, we will specifically explore the transcriptional upregulation of AQP4 in response to the inducer SFN. SFN stimulates transcriptional activation of the AQP4 gene [

The modeling of AQP4 transcription and translation reaction mechanisms. Nrf2 activation by SFN treatment upregulates AQP4 expression in the astrocyte cell.

SFN administration increases Nrf2 translocation into nucleus and Nrf2 binding to the aquaporin 4 gene promoter. As a result, the transcription rate is increased until the level of nuclear Nrf2 returns to normal due to Nrf2 nuclear export. In this model, translation kinetics is assumed to be a first order reaction in respect to mRNA concentration and all degradation kinetics are set to follow first order reaction.

Gene transcription is a stochastic process and the levels of transcription show great variability between single cells in a population [

In addition to transcriptional and translational reaction mechanisms, we also incorporate the transport phenomenon in the model. Two different transport mechanisms have been applied to describe movement in the cytoplasm; passive pure diffusion and active motor-protein driven convective transport. For the stochastic simulation of pure diffusion, we used fractional Gillespie multi-particle algorithm (fGMP) which is an approximation of the Gillespie method [

An information flow diagram for the stochastic simulation of reactions and intracellular transport is shown in

^{0} and ^{0} denote the degradation of each species in those propensity functions. Discrete numbers of all reaction events (R1–R6) are then sampled by _{j}_{b}_{j}_{j}_{j},_{tc}_{j}_{j},_{tl}_{j}_{j},_{ub}_{j}_{j},^{0}) = _{dr}_{j}_{j},^{0} = _{da}_{j}

Flowchart of the cell simulator.

Tau leaping method is an approximation method that leaps over many reaction events to approximate the exact stochastic simulation, while maintaining reasonable computing performance. Our discrete time approximation method uses a constant time interval for simulation, regardless of the number of events during each interval. The advantage of this method is that user-defined fixed time steps can be implemented. In discrete time approximation method, all events pertaining to the time interval are implemented before updating the values of the propensity function. The time step is adjusted according to the system complexity.

_{s} = 0.125 μm^{2}/s. It also depends on the mean distance to the neighboring subvolumes as expressed by the grid spacing parameter, λ, which has units length.

The event times for all volumes in the cellular domain are initially computed and simulation follows the increasing order of the least traveling time, _{min}_{min}_{i}_{i}_{i}

In order to integrate the spatial coordinates of the cytoskeletal network and that of the cellular subvolumes where diffusion and reactions occur, these two meshes exchange coordinate information in our cell simulator. One mesh serves as the domain for diffusion and reaction, and the other mesh serves to compute convective transport along the microtubules. In our algorithm, the mesh for the convective transport contains the microtubule structural identity and shares the mesh information with the volumetric cell mesh such as sub-volume id and absolute geometric information of all microtubule tracks. Therefore, a molecule associated with the microtubule track is allowed to propagate along the microtubule, updating its position along the microtubule. When the dissociation from the microtubule occurs, it communicates with the cell mesh to move that vesicle into that neighboring sub-volume.

To better characterize intracellular dynamics, the stochastic simulation for AQP4 synthesis was compared to a deterministic model. Averages of cell states obtained from repeated stochastic simulations were used for the side-by-side comparison with the continuous simulation results. The mathematical model for the intracellular kinetics using ordinary differential equations is described in Equations (11) to (14)
_{b}_{ub}_{tc}_{tl}_{dr}_{da}

The kinetic rates of our steady state model are determined from published AQP4 protein half-life. Since data on the stabilities of AQP4 transcripts are not available, we employ averaged properties of mRNA transcripts measured from more than four thousand mammalian genes [

We performed two sets of experiments in astrocyte cell culture; (

The steady state of the system describing AQP4 synthesis and transport was matched with existing data. AQP4 proteins have a known half-life of 24 h in the cell [

Kinetic parameters shown in Equations (11) to (14) were determined to arrive at physiological AQP4 protein and transcript levels. These levels agree with known data that on average approximately 2 copies of mRNA are synthesized per hour and 900 proteins from each mRNA are translated per hour [^{5} at steady state.

Kinetic rates of AQP4 synthesis.

Parameters | Values (s^{−1}) |
---|---|

Transcription rate with no SFN treatment | 5.55 × 10^{−4} |

Transcription rate with SFN treatment | 2.08 × 10^{−3} |

Translation rate | 0.25 |

Degradation rate of mRNA | 4.01 × 10^{−5} |

Degradation rate of AQP4 | 8.02 × 10^{−6} |

Upon sulforaphane (SFN) administration, the transcription factor Nrf2 becomes phosphorylated and translocates into the nucleus, and the transcription of the AQP4 gene is activated by Nrf2 in this model. The degree of induced gene activation in the cell model is derived from experimentally measured Nrf2 translocation in normal and SFN treated cells, see appendix for detailed procedures. Briefly, to determine the level of Nrf2 activation, levels of phosphorylated Nrf2 (pNrf2) within the nucleus were quantified with immunofluorescence images obtained under controlled staining and imaging conditions. After continuous SFN treatment, the fluorescence levels of pNrf2 increased by about 63% compared to control at 15 min, and returned to normal levels by the 9-h time point.

Our observation of rapid Nrf2 nuclear translocation is in agreement with findings by Jain

The dynamics of the SFN-induced AQP4 upregulation is predicted with the cell simulator. A discrete event simulation was performed with the approximated Gillespie method, using a discretized time step of ∆

Stochastic simulation results of AQP4 transcripts and proteins after sulforaphane exposure. Each red solid line represents one stochastic realization and dotted black line is the solution of the deterministic simulation. The upregulation of AQP4 normalized to control was 1.48 ± 0.11 at 9 h and 1.68 ± 0.20 at 18 h.

In brain astrocytes, AQP4 channels are highly expressed in the endfeet [

Finally, we demonstrate a qualitative comparison of the spatial AQP4 distribution in a single astrocyte cell with our simulated results as shown in

However, the number of AQP4 containing vesicles or AQP4 tetramers concentrated at a single endfoot cannot be quantified with the immunofluorescence technique that was employed in this study due to limitations in resolution. In future studies, a quantitative validation of AQP4 spatial polarization at the astrocytic endfeet in the model could be performed with techniques such as freeze fracture electron microscopy [

AQP4 trafficking in primary cultured astrocytes (

Western blot (WB) was performed to quantify the upregulation of AQP4 by 13 µM of SFN. For SDS-PAGE gel electrophoresis, cell lysates were prepared after 9 and 18 h of continuous SFN exposure. The results show that AQP4 expression was 1.48 and 1.68 fold higher than control after 9 and 18 h of SFN exposure as shown in

The comparison between measurement and simulated AQP4 upregulation is shown in

Western blotting shows that SFN induced a 48% upregulation of AQP4 after 9 h of continuous exposure, and 68% upregulation after 18 h compared to control (

Many different molecular biological assays and biochemical techniques can be used to investigate intracellular process encompassing events starting at gene activation to the final expression of its encoded protein. These datasets provide time profiles of gene activation, transcripts, or protein levels and its spatial distribution pattern in a cell or in a population of cells. In order to derive a complete understanding of the dynamics of protein expression in a particular system, however, it is difficult to quantitatively measure every species involved in the biogenesis of a protein at every step in a time-dependent fashion due to limitations in time and resources. Oftentimes, data on relative expression levels allows the postulation of transcription and translation dynamics of the system, but these hidden kinetics cannot be extracted without using quantitative tools. Furthermore, time-dependent protein localization in specific cellular domains observed by fluorescence microscopy can be incorporated to derive the kinetics of trafficking and expression, after the transcription and translation step. Our case study derived additional insights about the unknown aspects of the chosen system: gene activation status and dynamic transcript levels, based on experimental observations on the transcription factor translocation and the target protein upregulation.

The single cell simulator provides a platform for performing quantitative analysis of cellular events using and integrating biological data from various common molecular biology techniques. To simulate trafficking events, the microtubule cytoskeleton was generated by a novel growth algorithm. In addition, the event-based stochastic simulation technique allows the tracking of individual molecules at any point in time. The approximated Gillespie algorithm reduced the computational cost and captured the discrete and stochastic nature of transcriptional and translational events inside a single cell. Conversely, the average of repeated stochastic simulations is suitable for representing the behavior of an entire cell population. Finally, we used a deterministic model to validate the averaged trajectories from our stochastic computations with good agreement. The analysis of cellular events

_{b}

DNA activation rate

_{ub}

DNA inactivation rate

_{tc}

mRNA transcription rate

_{tl}

AQP4 translation rate

_{dr}

mRNA decay rate

_{da}

AQP4 decay rate

activated DNA molecule

The authors gratefully acknowledge Simon Alford for generating the images of primary astrocytes with confocal microscopy.

The authors declare no conflict of interest.

Nrf2 is a putative TF of the AQP4 gene [

Rat primary cortical astrocytes were purchased from Lonza. Cells were grown at 37 degrees Celsius with 5% CO_{2}. Astrocyte growth media (Gibco, Madison, WI, USA) contains 10% FBS supplemented with 1% penicillin/streptomycin and 1:500 amphotericin B. Growth media was changed twice a week and cells were passaged when confluent.

To quantify AQP4 upregulation by SFN, cell lysates were prepared by lysing with RIPA buffer after 9 h and 18 h of continuous SFN treatment at 13 µM. Bradford assay was performed to ensure equal loading of proteins. SDS-PAGE gel electrophoresis was performed on a TetraCell Mini system (BioRad, Hercules, CA, USA) using AnykD polyacrylamide gels (Biorad, Hercules, CA, USA). After transfer and blocking in bovine serum albumin, membrane was incubated with mouse anti-AQP4 primary antibody (Abcam, Cambridge, MA, USA) overnight. On the next day, secondary antibody (anti-mouse tagged with HRP) was applied for 1.5 h followed by washing steps. Membrane was imaged with BioRad Chemiluminescence detection system (BioRad, Hercules, CA, USA). Large molecular weight aggregates were excluded from the quantitative analysis. Densitometry was performed with Image J [

Diffusion of GFP-tagged fusion proteins inside distinct cellular compartments (ER, Golgi, cytoplasm) have been measured, and there is a wide range of diffusion rates depending on the size of the molecule and the specific cellular compartment. For example, diffusion of GFP (which is very small in size) inside the cytoplasm is 25 µm^{2}/s [^{2}/s [^{2}/s [^{2}/s in our model, which is a reasonable estimate for computing the diffusion of molecules with a larger size.