In this work, we used E-Cell, a software package aiming at large-scale modeling with full object-oriented modeling support, to analyze the 70kDa heat shock protein (Hsp70) chaperone mediated protein folding. We analyzed the kinetic characteristics of this chaperone system during folding of an unfolded protein using computer simulations. Our simulation results are consistent with reported laboratory experiments and support the kinetic partitioning hypothesis. Our model suggests that although the DnaK chaperone system is robust in assisting protein folding, this robustness is limited by the availability of ATP. Based on this model, we also discuss why object-oriented modeling is needed to reduce the complexity of large-scale biochemical models.

Introduction

Although the entire information for the precise three-dimensional structure of a protein is encoded in its amino acid sequence, in vivo many proteins depend on the assistance of molecular chaperones such as Hsp70 (DnaK) and Hsp60 (GroEL) heat shock proteins for folding from a nascent or denatured state into their correct structure.^1,2 It is known that in Escherichia coli the DnaK, DnaJ and GrpE chaperone machinery can efficiently repair misfolded Photinus pyralis luciferase both in vivo and in vitro but cannot protect it from heat induced unfolding.³ Experimental findings suggest that this refolding process is achieved through ATP-dependent interaction between the DnaK chaperone and the substrate protein or peptides.⁴ DnaJ and GrpE function as regulators in this system by stimulating DnaK's ATP hydrolysis activity and subsequent nucleotide exchange.^5-8

The kinetics of the DnaK chaperone system has been studied extensively.^9,10 Different mechanisms have been suggested to explain the steps in chaperone action. The mechanism suggested by Schröder et al³ has been widely accepted. In this proposed mechanism, an unfolded protein substrate (e.g., Photinus pyralis luciferase) first associates with DnaJ, which will present it to DnaK.ATP and induce the formation of a trimeric DnaK.ATP.DnaJ.substrate complex. DnaJ and substrate synergistically stimulate ATP hydrolysis by DnaK and thereby trigger the transition of DnaK from the ATP state with low affinities for substrates to the high-affinity ADP state. GrpE will bind to the latter complex and catalyze the release of ADP. Subsequent ATP binding induces conformational changes in the ATPase domain and substrate binding domain leading to a rapid dissociation of GrpE and substrate from the complex. These steps form a cycle of the DnaK-assistant folding. With enough ATP and all the chaperone molecules, after many cycles, the substrate can be refolded back to its active state (Fig. 1, see also ref. 3). Here we describe a kinetic model for DnaK chaperone action in protein refolding based on Figure 1. The rate constants were derived from literature or completed by our experiments. Our model is shown to simulate correctly the behavior of E. coli DnaK chaperone action. The kinetic partition hypothesis proposed for protein refolding, the sensitivity of refolding productivity to alterations in activity and concentration of the chaperones and ATP consumption are discussed.

Figure 1

Kinetic model of DnaK chaperone system in protein refolding. Arrows indicate the reaction direction. S is an abbreviation of substrate. GrpE2 is for the dimmer form of GrpE, which is the active form. Pi is for phosphate acid. Dot (·) in between (more...)

Materials and Methods

Reaction and Parameters

The model is based on the rate equations derived from the kinetic model in Figure 1. All the reactions and parameters used in this computer model were either based on published literatures or measured in the M.P.M. lab. For each substrate protein passing through a cycle of refolding process, a probability is assigned for it to be fully refolded. Protein aggregation is not included in our model. This is because of: (1) Lack of quantified data on aggregation; (2) The model is developed to test the property of the DnaK chaperone system at physiologically optimal growing temperatures, where the probability of protein aggregation is small. Please refer to Table1 for a reaction list for DnaK chaperone kinetic model.

Table 1

Reaction list for DnaK chaperone kinetic model.

Simulation and Plotting

The simulation was based on an improved version of Gillespie's exact stochastic simulation algorithm²¹ by Gibson and Bruck,²² as implemented in E-Cell v3, an open source computer software package for large scale cellular events simulation,^22,24 developed at Keio University (www.e-cell. org). The simulation results were plotted by using GnuPlot (www.gnuplot.info). The power fitting in Figure 5 was done with the Microsoft Excel program.

Figure 5

ATP consumption and the probability of refolding. After more than 95% of the substrate was fully refolded, ratios of ATP consumed were calculated and divided by the number of transformed substrates. Probability is the probability value of one substrate (more...)

Model Validation

To validate our model, we compared our model results with those published in.³ As shown in Figure 2, with a refolding probability of 1.16% in a single cycle, results from our model are in onsistent with laboratory results. A probability value of 1.96% could bring a result with similar dynamics with faster refolding rate.

Figure 2

Model validation. In vitro experiment result as published in. With a refolding probability of 1.16%, our result fits well with their report. Different probability values can result in different refolding speed, although other parameters are the same. (more...)

Results

Partition of the Substrate Binding

Based on in vitro experiments, it has been proposed that the substrate binding to chaperone follows a kinetic partitioning.⁹ In the fast phase, which may take seconds to minutes, more than half of the substrates are bound by chaperones. Later, in the slow phase, the rest, about 50% of the substrates, slowly associate with the chaperones. Our model accurately reproduced this phenomenon when we introduced 1000 molecules of unfolded substrates into the system and simulated the generation of refolded substrates. After about 30 minutes, more than 99% of the substrates were fully refolded (Fig. 3). This result shows the kinetic partitioning is an inherent property of this network with the parameter set we used. It also indicates that the function of DnaK.ATP being able to combine with substrate directly may provide a buffer of the subsequent refolding reactions.

Figure 3

Partition of substrate binding.

Robustness Analysis

Robustness can be defined as the insensitivity to changes in variables. Here we tested the robustness of the DnaK chaperone system by reducing the initial values for the numbers of DnaK, DnaJ and GrpE molecules by half. Figure 4 shows that after a 50% reduction of the amounts of these chaperone molecules, the system is still able to maintain its behavior for refolding the substrate, which indicates that the chaperone system is robustly designed. Among the three chaperone molecules tested, a 50% reduction of DnaJ and GrpE had the strongest impact on the refolding process and no significant differences could be found in the results when DnaK and ATP was varied. This is surprising and contrasts in vitro observations. We will discuss this matter below.

Figure 4

Robustness with respect to initial values.

Discussion

Based on published literature and our experimental results, we developed a kinetic model for analyzing the DnaK chaperone system in folding de novo synthesized polypeptides or refolding of unfolded proteins at ambient temperatures. Using this model, we analyzed the kinetic partitioning found in DnaK substrate binding reactions and tested the sensitivities of the system's function with respect to DnaK, DnaJ and GrpE. Our model accurately represents the laboratory findings, except the effects of decreased DnaK concentration and activity. The results of the simulations demonstrate that GrpE is the most sensitive component (data not shown) among the three chaperones, which rationales the potential function of GrpE as a thermosensor.¹¹

The inconsistency of the effect of DnaK concentration and activity between our model and laboratory reports in¹⁰ can have at least two alternative explanations. 1) In our model, we have not included substrate aggregation. If luciferase aggregation is considered, the system will be much more sensitive to the DnaK concentration. This is because DnaK binding of unfolded luciferase will compete the self-aggregation of luciferase. Thus, a part of the high sensitivity of DnaK concentration found in¹⁰ may be due to the aggregation prevention function. 2) In our model we have not considered the possibility of more than one molecule of DnaK binding simultaneously to a single substrate, since to our knowledge there is no experimental evidence published on this issue so far. However, we hypothesize that this is a possible mechanism in vivo to enhance the refolding process. Such a mechanism also would be more sensitive to a reduction of concentration and activity of DnaK. Further laboratory experiments are needed to validate this hypothesis. Protein aggregation and synergistic action of several DnaK molecules during the refolding of a single substrate shall be implemented into future versions of our model when experimental evidence will allow an estimation of the kinetic parameters involved.

One question of debate concerning the DnaK-chaperone cycle was the dissociation of DnaJ. Based on the fact that DnaJ is only 1/10th to 1/30th as abundant as DnaK in vivo¹⁴ and can act substoichiometrically in vitro^15,8 it was assumed that DnaJ leaves the cycle just after the transfer of the substrate onto DnaK and before GrpE binds to the complex.⁴ This scenario was chosen for our model. The elucidation of the binding sites for GrpE in the cocrystal structure with DnaK¹⁶ and for DnaJ through genetic and biochemical analyses^17,18 indicated that both DnaJ and GrpE could eventually bind at the same time to DnaK and possibly to the DnaK-substrate complex. We therefore asked the question whether the refolding efficacy for the substrate would change when DnaJ leaves the cycle together with GrpE upon binding of ATP instead of before the binding of GrpE. However, the results did not change when only the exit point for DnaJ was varied (data not shown). Therefore, despite the substoichiometric concentration of DnaJ, the actual exit point of DnaJ is not critical for the refolding efficacy as long as a quarternary complex of DnaK with DnaJ, substrate and GrpE does not change the dissociation kinetics significantly or has any other additional effect on the refolding probability of the substrate.

Robustness, i.e., buffering relatively large alterations in system parameters, is a natural property of many biological systems and should be expected for the Hsp70 chaperone system as well. Under stress situations, the availability and activity of Hsp70 chaperones may be reduced and it is important to know how the chaperone function is affected under these conditions. Such questions are generally difficult to address experimentally (see also 14). In robustness tests of our model we found that this chaperone system is robustly built to refold proteins. A 50% reduction in the concentration (Fig. 4) did not affect the behavior of refolding dramatically. In the absence of side reactions such as aggregation, the unfolded substrates, once they enter this pathway, will complete their destiny towards refolding. Within the range tested, fluctuations in concentration or activity of the chaperones only delayed the refolding process and did not change the overall behavior. Only severe activity reductions caused significantly longer delays. Such a robust design provides the fundamentals of the heat shock response,¹⁹ where the DnaK chaperone system assists cells to survive temperature increases by refolding heat denatured proteins. Some of this apparent robustness may be due to the exclusion of side reactions such as aggregation and will be investigated in future implementations of the model. Nevertheless, our robustness analysis emphasizes that exclusion or reversion of side reactions are major issues for the cells under stress conditions since the actual refolding reaction can cope with relatively large fluctuations in chaperone abundance and activity.

The nonlinear relationship between ATP consumption and the refolding probability (Fig. 5) is also interesting. It is known that in ΔrpoH mutants, which lack the heat shock transcription factor and therefore have low levels of all major cytosolic proteases and chaperones except GroEL, 5-10% and 20-30% of all total proteins aggregated at 30°C and 42°C, respectively. The aggregates contained 350-400 protein species.²⁰ Since the DnaK system is beside GroEL the only chaperone system in E. coli that is able to refold proteins to their native state, these protein species must be substrates for DnaK under normal conditions. These proteins may cover a wide range of probabilities of refolding, which is most likely determined by their sequence and fold. Although ATP has under optimal growth conditions a relatively high total concentration (3 mM), in stress situations the effective free concentration of ATP may nevertheless be insufficient to meet the challenge of refolding hundreds of different molecules simultaneously. Thus, we think the cellular amount of ATP may actually limit the robustness of the Hsp70 chaperone system in its protein folding function.

Currently, the complexity in the model is limited because we are focusing on the situation where only one substrate species exists. But if we are going to simulate the refolding of 100 different substrates at the same time, the complexity of model construction easily goes up and becomes hard to manage. One possible solution is to borrow the idea of object-orientation from computer science. Consider that if all the protein species have the inherent property of refolding and aggregation, etc (which is true in nature) inside the simulation software when we construct a large-scale model, we can just send the protein a message “fold” and it would find its way towards the folding process. In this way, we can replace the thousands of reactions with just 100 messages. Thus, object-oriented modeling is a prominent solution for complex biological models.

Acknowledgments

The contents of this chapter were based on a previous publication by the same authors.¹ A more detailed model based on this work can be found in.²⁶ This work is supported in part by grants from the Ministry of Education, Culture, Sports, Science and Technology of Japan(Leading Project for Biosimulation, The 21st Century COE Program and Grant-in-Aid for Scientific Research on Priority Areas) and a grant from CREST, JST. This work was also supported by the Deutsche Forschungsgemeinschaft (SFB638 to M.P.M.).

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Footnotes

1

Bin Hu, Matthias p. Mayer and Masaru Tomita (2005) HSP-mediated Prote in Refolding in the E-Cell, in The Proceedings of The 9 World Multi-Conference on Systemics, Cybernetics and Informatics VIII, Callaos N et al Ed. 377-382.