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Conceived and designed the experiments: MW XY CC. Performed the experiments: MW XY. Analyzed the data: MW XY. Contributed reagents/materials/analysis tools: CC. Wrote the paper: MW XY CC.

The authors have declared that no competing interests exist.

A major challenge in systems biology is to develop a detailed dynamic understanding of the functions and behaviors in a particular cellular system, which depends on the elements and their inter-relationships in a specific network. Computational modeling plays an integral part in the study of network dynamics and uncovering the underlying mechanisms. Here we proposed a systematic approach that incorporates discrete dynamic modeling and experimental data to reconstruct a phenotype-specific network of cell signaling. A dynamic analysis of the insulin signaling system in liver cells provides a proof-of-concept application of the proposed methodology. Our group recently identified that double-stranded RNA-dependent protein kinase (PKR) plays an important role in the insulin signaling network. The dynamic behavior of the insulin signaling network is tuned by a variety of feedback pathways, many of which have the potential to cross talk with PKR. Given the complexity of insulin signaling, it is inefficient to experimentally test all possible interactions in the network to determine which pathways are functioning in our cell system. Our discrete dynamic model provides an

A major challenge in current molecular biology is to understand the dynamic behavior of biological systems. Biological processes consist of many interacting components, exceeding the human capacity to systematically analyze them, thus requiring methods to reduce the complexity and thereby enhance their accessibility

Here we proposed a systematic, dynamic analysis approach that reconstructs a phenotype-specific network of cell signaling. As an example, insulin signaling, a well-studied and complicated signaling network, in mammalian cells is composed of branched downstream signaling pathways and various feedback mechanisms, which could benefit from modeling. In a separate study, our group identified the involvement of a novel player, PKR, in the insulin signaling network of HepG2 cells

A dynamic model that can efficiently integrate the literature knowledge and experimental data

Discrete dynamic or Boolean network modeling applies Boolean algebra to obtain a qualitative, discrete representation of the biological system

Discrete modeling has uncovered many important dynamic features in biological systems. The dynamic model developed by Li et al

Insulin is one of the major hormones controlling the complex hepatic metabolic responses. Insulin binds to the insulin receptor tyrosine kinase (IR), which recruits and phosphorylates the insulin receptor substrate (IRS) proteins at tyrosine residues

Insulin signaling is tuned at the IRS level, by a large number of regulators

Given the complexity of insulin signaling, it is inefficient to experimentally test all possible interactions in the network to confirm their functionality. Thus, we proposed a novel discrete dynamic network model that integrates potential interactions and components of insulin signaling and its feedbacks in HepG2 cells. Our simulations provided patterns of dynamic activation of each component and generated testable hypothesis on the response of the network upon perturbation, which were used to direct the experiments.

The dynamic behaviors are determined primarily by the network architecture. This assumption is based on the observation that biological systems tend to maintain their functionality despite environmental and intrinsic noise that cause fluctuations in their protein/RNA levels or reaction rates

In Boolean networks, the binary on/off representations of mRNA or protein level, and the logical functions for the interactions can be directly derived from qualitative experimental data of activating/inhibitory role of the various elements in the network. Since the switching behavior (i.e., activation/inhibition) is quick in many signal transduction processes, it is reasonable to approximate the actual continuous process of the state of the network components with discrete levels

Given that modularity is a design principle of biological systems

The signaling network can be formalized in terms of an oriented graph, where the vertices represent the elementary components involved in the process and the arcs describe the regulatory interactions between those components. In the signaling network, each directed arc reflects the direction of information flow from the source vertex to its target in the signal transduction, and is labeled with a positive or negative sign which defines activation or inhibition, respectively. The network in

Each arc is assigned an attribute—either activation or inhibition.

We represent the cell variation by the fluctuations in their initial protein activity levels, which we assume to be a normal distribution around a control state for each protein. We associate each vertex in the signaling network with a discrete variable, which has three states representing the activity of the protein----0: lower than control, 1: the control state, 2: higher than control. Thus by definition every component starts at the “control” state in the absence of insulin stimulation. We use a discrete model with variables and transition rules (see below) to represent the dynamic profile of a cell and we sample a large population of cells. We choose an initial state for each regulatory component from a distribution centered at the “control” state, which reflects the cell-to-cell variation in the protein levels or activities. The variation captured by the distribution represents the degree of stochastic noise in the protein activities.

Constraints are assigned to components to mimic the perturbations on the network. If a protein is constantly inhibited (in an experiment), we restrain the state of its corresponding variable in the model to be always in either 0 (lower than control) or 1 (control state). Since we apply a distribution on the initial states of the protein activity for describing the cell-to-cell variation, we can use the variation over the distribution of initial states to describe the degree of noise that causes the cell-to-cell variation in the protein activities. Variation in the cellular protein activity can lead to cell-to-cell variation

We define transition rules based on the activation/inhibition attributes on the arcs in the signaling network. Two operations: shift up and shift down adapted from the triple logic are applied in the model (see

Shift up | Shift down | ||

Current-State | Updated-State | Current-State | Updated-State |

0 | 1 | 0 | 0 |

1 | 2 | 1 | 0 |

2 | 2 | 2 | 1 |

In the simulation each run starts with its own set of randomly generated initial states and a simulation result represents the dynamic profile of a single cell in the population. By assuming that cells response independently to a signal, we can simulate a large number of independent runs to mimic a population effect and measure the average evolving profile for the population (see

Time is modeled by regular intervals called time-steps. Since most components in our network are kinases or phosphotases, and most reactions are protein phosphorylation and dephosphorylation, we assume that the duration of the activation/inhibitions and the decay processes in the signaling transduction are comparable and approximated by one time-step. Since the reaction rates may be different from cell to cell even for the same interaction, we apply asynchronous updating of the state, which is realized immediately, rather than renewing every variable simultaneously at each time-step. Thus, the relative rates of the different reactions can be specified by the ordering of the update, which implies that, although the response may be similar, the rate of response varies from cell to cell. The rules for an asynchronous algorithm can be written as:

Since we embed the uncertainties in the signaling process by applying random initial states and the asynchronous updating, we need a population large enough to obtain a stable dynamic profile. As shown in the graph (

The dynamic model was implemented by custom MATLAB code, with a random-order asynchronous updating algorithm, which updates each component, one by one, in a randomized order at each time-step, in each cell. The order in which the components are updated is randomly chosen from a uniform distribution over all possible permutations, to achieve random timing for each regulatory interaction. As an example, let us consider a single interaction: component A activates component B. Suppose A and B are both in 1, the control state, but at the next time-step a stimulus activates A and shifts its state to 2. If applying synchronous updating, the state of B at this “next time-step” is 1 (since it is based on the current state of A, which is 1), so B will be activated in the following step, thus the time required for the activation from A to B is fixed to one time-step. In asynchronous updating, the state of B on the next time-step is either 1 or 2 depending on whether the updating of B is prior to the updating of A, which is randomly chosen in a cell, thus the timing of the activation is variable. The state-change process of this example is shown in

Human hepatoblastoma cells (HepG2/C3A) were cultured in Dulbecco's Modified Eagle Medium (DMEM) (Invitrogen, Carlsbad, CA) with 10% fetal bovine serum (FBS) (Biomeda Corp, Foster City, CA) and penicillin-streptomycin (penicillin: 10,000 U/ml, streptomycin: 10,000 µg/ml) (Invitrogen, Carlsbad, CA). Freshly trypsinized HepG2 cells were suspended at 5×105 cells/ml in standard HepG2 culture medium and seeded at a density of 106 cells per well in standard six-well tissue culture plates. After seeding, the cells were incubated at 37°C in a 90% air/10% CO2 atmosphere, and two milliliter of fresh medium was supplied every other day to the cultures after removal of the supernatant. The HepG2 cells were cultured in standard medium for 5–6 days to achieve 90% confluent before any treatment. Human insulin was purchased from Sigma-Aldrich (St. Louis, MO), okadaic acid (OA), IKK inhibitor (SC-514), JNK inhibitor (SP600125), and ERK inhibitor (PD 98059) from EMD Biosciences (San Diego, CA).

Human insulin was stocked in HEPES buffer, which was therefore used in controls for all the experiments with insulin treatment. We treated the cells with insulin at concentrations lower than 1 nM to mimic the physiological concentrations

HepG2 cells were lysed as described previously

We collected and integrated the literature information on insulin signaling with emphasis on the different feedback pathways and crosstalk with PKR and built a consensus regulatory network that contained all the components and potential interactions (see

Each simulation represented a single cell and we simulated a large population to obtain an average pattern of the dynamic activation of each component in the network, with or without insulin stimulation. We compared the simulated and the experimental profiles of IRS tyrosine, IRS serine, and PKR phosphorylation levels.

The simulation result suggests plausible dynamic profiles of the interacting network upon insulin stimulation, and based upon the components and interactions potentially involved in our particular cell system (

A) Model simulations with or without insulin stimulation. The simulation is on the initial model including the potential interactions and components from the literature and our experiments. The interactions emphasized are the level of IRS serine phosphorylation (IRSS), IRS tyrosine phosphorylation (IRST), and the PKR phosphorylation. B) Time series of the PKR and IRS phosphorylation upon insulin stimulation at time 0. HepG2 cells were exposed to 1 nM of insulin for 5, 10, 15, 30, or 60 minutes. After treatment, the cells were harvested, and western blot analysis was performed to detect the total and phosphorylated levels of PKR and IRS1

We designed several western blot experiments to measure the changes in IRS tyrosine, IRS serine, and PKR phosphorylation levels in response to insulin in liver cells. The experimental results (

However, from the experiments, we observed that the PKR phosphorylation decreases with the IRS tyrosine phosphorylation, but rather than remaining at a constant, low level as predicted by our model, PKR phosphorylation actually increases 15 minutes after insulin stimulation (

We then examined the contribution of each regulatory interaction through

A)

It has been shown in different systems, but not in HepG2 cells, that PP2A can induce the dephosphorylation of Akt at amino acid residue Thr308, and thereby suppress the activity of Akt

The

In order to elucidate the contribution of the different feedback pathways, we evaluated topologies that contained only two of the three pathways, JNK+IKK, ERK+JNK, and ERK+IKK, in addition to the original topology containing all the ERK, JNK, and IKK feedbacks (

Each column represents a different architecture containing. A) Original ERK+JNK+IKK feedback pathways, B) Only with IKK+JNK feedback pathways, C) Only with ERK+JNK feedback pathways, and D) Only with ERK+IKK feedback pathways. Row 1 the model simulation is performed without any perturbations. Row 2 the model simulation is performed on IKK inhibition. Row 3 the model simulation is performed on JNK inhibition. Row 4 the model simulation is performed on ERK inhibition. E) Western blot: effects of JNK, IKK and ERK inhibitors on the phosphorylation of IRS1 upon simulation with 1 nM insulin for 15 mins.

The experimental result shows the JNK inhibitor significantly down-regulated the IRS serine phosphorylation and up-regulated the IRS tyrosine phosphorylation, as compared with the normal, unperturbed insulin stimulated network. Therefore, the model without the JNK feedback (

Among the three single-elimination models, removing the ERK pathway (JNK+IKK) provided the most significant change in the dynamic profiles of IRS phosphorylation upon JNK inhibition as compared with that of the IKK inhibition (

To further evaluate which pathways are essential in the dynamic model, we simulated the case of PKR over-expression with the network topology model without the ERK feedback to serine phosphorylation of IRS.

A)

The direct interaction between PP2A and AKT is removed. The ERK feedback is removed, together with the downstream factors in the ERK pathway that do not have an effect on the regulation of insulin signaling.

We used the distribution of initial states to describe the cell-to-cell variation of protein activity levels due to environmental or endogenous noise. The variation over the distribution of initial states represents the degree of noise. Multiple simulations were run on the architecture consisting of the essential pathways, under varying levels of noise, with and without insulin stimulation. The simulation results (

In row 1 the noise is represented by the variations over the distribution of the initial states of each component. Simulation without or with stimulus are shown in row 2 and row 3, respectively. In column 1, 70% of the initial protein activity levels are at the control state, indicating minor noise; in column 2, 50% are at the control state; and in column 2, the components are equally likely to be assigned to any of the three states, indicating a higher level of noise. Simulations are based upon the essential pathway model that excludes the ERK pathway. Colors on the distribution: light grey: control state (1), dark grey: higher than control (2), black: lower than control (0).

We further simulated the perturbation models under a different noise level (e.g., uniform distributed initial states) and found the dynamic profiles to be robust (

We enhanced the discrete modeling approach by extending the on/off logic of traditional discrete Boolean models to a three-level logic model with “high, control, low”. Also, rather than assigning ad-hoc absolute values as initial states, for each component we defined its state prior to stimulation by a distribution centered on the “control state” and compared the effect of the perturbation with the control state, and thereby avoiding the problem of defining the initial state in traditional Boolean network models.

In contrast to the traditional Boolean models that apply one model to describe the whole process, we apply an individual model to every single cell with each cell having a distinct initial state from a distribution centered on the control state. The simulation produces a response by obtaining the average effect of a large group of cells. This modeling approach takes into account the population effect and cell-to-cell variation, such that the resultant model can capture the deviations or alterations from the control state that is conceptually consistent with the western blot experiments.

Discrete dynamic models do not represent or capture the exact timing, but the resultant dynamic patterns provide a dynamic profile evolving with “time-steps”, which may contain artifacts due to the network reconstruction and the updating rules at each time step. Previous biological Boolean network models were simulated in such a way that all the components changed their states simultaneously by one unit of time, based on the assumption that every reaction in the network takes exactly one unit of time in the signaling process. However, in a real biological system, the reactions are not homogenous, different reactions may have different rates, thus “synchronous” updating may not be appropriate. Klemm et al.

Asynchronous updating has been suggested to reduce the artifacts due to the assumption of uniformity in all the reactions arising from synchronous updating

Even with asynchronous updating, there still may be artifacts that arise due to the discrete timing, such as small fluctuations. For example, a simple regulatory network with only three components is shown in the

Discrete models are based solely on the qualitative relationships of the network elements, and thus may lose certain subtle dynamics. It has been shown that in some cases certain kinetic parameters are essential for a system's behavior (e.g., the reaction constants of phosphatases determine the response time and duration under weak stimulation in the MAPK kinase cascade model in

Since our insulin signaling network is somewhat homogenous with only protein phosphorylations/dephosphorylations, our model treats every interaction and feedback the same, and does not take into account the different strength or time scale of the interactions and feedbacks, which are possible in real systems, especially in heterogeneous networks with both protein interactions and transcription regulations. Nevertheless, as long as the quantitative experimental data are available that captures such differences, we can further adjust the model to integrate such information, by increasing the discrete levels and applying more subtle rules. For example, one can apply longer updating time for transcription regulations than for protein interactions.

The advantage of discrete modeling is that only qualitative information is required. Kinetic modeling (and stochastic modeling) requires detailed kinetic parameters and initial concentrations for each component in the network, which is usually not available for all the components in the network. Without the need for kinetic parameters, the discrete modeling approach can model and simulate larger networks, given the abundant qualitative interaction maps of biological systems that are available.

Discrete dynamic modeling represents a higher abstraction of biological network as compared with kinetic modeling, and perturbations on the structure of the network are more easily simulated with the discrete models. Kinetic approaches usually perturb parameters of a system rather than the structure, because a different set of kinetic parameters (and even different equations) may be required if the network structure is changed. Therefore, it is very difficult to obtain those parameters for the perturbed system, despite the availability of experimental data.

As a simplified qualitative model, discrete dynamic model has finite system-states, thus one can analyze the design principle of the network structure by exploring all the possible states to find the stationary ones

In conclusion, our novel modeling approach provides a systematic description of the biological process, which enables testable predictions that can serve as working hypothesis for experimental evaluation. By combing modeling and experiments, we can develop a hypothesis-driven framework that can be iteratively refined to enhance our understanding of the insulin signaling network dynamics in the context of a particular cell system. Such a framework that relies solely on the network architecture can be easily extended to other dynamic network systems and serves as a basis to guide model-based experiments and potentially more detailed inquiry into the regulatory mechanisms of biological networks.

With a small population the dynamic profile varies significantly in the different simulations due to the embedded randomness, whereas in a large population the profile is stable.

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An example of the implementation of synchronous and asynchronous updating for a cell.

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In silico knock-out. For each subgraph, an interaction is deleted from the model and the simulation is performed on the knock-out model. The interactions being knocked out is labeled at the top of each subgraph in the form of “no”-regulator-target. Red line: IRS serine phosphorylation, green line: IRS tyrosine phosphorylation, blue line: PKR phosphorylation.

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A local positive feedback module from AKT to PP2A

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The effects of ERK silencing or inhibition on the phosphorylation of IRS1 in PKR over-expressed cells. (A) Reverse transfection of suspended HepG2 cells were performed with scrambled siRNA (NC) or siRNAs of ERK1 and ERK2 together for 24 hours and the transfected cells were cultured in regular media for another 24 hours. Next, the forward transfection of empty vector pCMV6-XL5 (pCMV6) or plasmid containing PKR cDNA sequence (pCMV6-hPKR) was performed, followed by the treatment of insulin (0.5 nM) for 15 minutes. After treatments, cells were then harvested and western blot analysis was performed to detect the protein level of ERK, and total and phosphorylated levels of PKR and IRS1. (B) In HepG2 cells, the forward transfection of empty vector pCMV6-XL5 (pCMV6) or plasmid containing PKR cDNA sequence (pCMV6-hPKR) was performed and the cells were then treated with the pharmaceutical inhibitor of ERK, PD98059 (PD98059, 50 uM) or DMSO, vehicle of PD98059, for 1 hour, followed by the treatment of insulin (0.5 nM) for 15 minutes. After treatments, cells were then harvested and western blot analysis was performed to detect the total and phosphorylated levels of IRS1 and PKR.

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Robust dynamics against noise. The noise is represented by the variation in the distribution of initial states for each component. Colors on the distribution: light grey: control state (1), dark grey: higher than control (2), black: lower than control (0). Perturbations and simulations are based upon the essential pathway model that excludes the ERK pathway, with uniform distributed initial state for each component.

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A small regulatory network with catalytic activation/inhibition. S is the stimulator (input) of the system. Upon stimulation of S, A activates B. There is a negative feedback from B to A. Simulation results of a kinetic model, a qualitative (Boolean) model with asynchronous or synchronous updating are shown.

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