(636a) Eukaryotic Nonlinear Dynamics and Intracellular Heat-Shock Bifurcations

The heat-shock response mechanism (Lindquist, 1986) is a vital, ubiquitous molecular reaction  ensuring eukaryotic homeostasis; it has evolved as an adaptation and survival mechanism to the proteotoxicity resulting from various classes of non-native (misfolded and damaged) proteins. Such non-native protein species frequently result in dangerous generation of protein aggregates. To eliminate abnormal proteins, cells employ a complicated machinery of molecular chaperones: the latter facilitate the refolding or degradation of misfolded polypeptides, prevent protein aggregation and assist in the formation of aggresome, to which small aggregates are transported. Protein folding and aggregation is studied by neurochemists and molecular neurobiologists, who increasingly recognize them as responsible for a number of serious neurodegenerative diseases: Alzheimer's, Parkinson's and Huntington's diseases and Amyotrophic Lateral Sclerosis affect millions and have a dramatic societal impact (Muchowski & Wacker, 2005; Szilágyi et al., 2007).

Heat-shock proteins (HSPs) function as molecular chaperones and are central to this regulation. When sensing stress signals (e.g. elevated temperatures, toxic molecules, oxidants, heavy metals) cells respond by transient molecular chaperone overexpression, to meet the stress at high levels. Chaperones recognize and bind onto exposed hydrophobic patches on unfolded polypeptides and conformational intermediates and assist towards either proper refolding or orderly degradation.

The function of the heat-shock transcription factor-1 (HSF-1), central to the heat-shock response mechanism, has been identified (Morimoto, 1993) and studied (Kline & Morimoto, 1997) early. Detailed network representations and dynamic mathematical models (Rieger et al., 2005) have been derived to identify species and study their concentrations in the heat-shock response cycle. Parameter estimation and sensitivity analysis by Rieger et al. (2005) provided additional insight regarding the relative importance of many kinetic parameters in the HSF-1 expression network.

This paper contains a rigorous structural analysis of the dynamic mathematical model (Rieger et al., 2005), illustrates the critical nonlinearities dominating the heat-shock network and presents a dynamic simulation study with respect to an uncertain, temperature-dependent parameter (β_m,k). A bifurcation analysis (Seydel, 1994) is performed via MATCONT (Dhooge et al., 2003) after demonstrating steady state multiplicity, to understand the hitherto unexplored effect of β_m,k on system response and state trajectories, for a multitude of biologically relevant equilibrium points. Choice of relevant initial conditions is crucial, even under mild nonlinearity (Gerogiorgis, 2009). Furthermore, an exponential correlation has been used to model the nonlinear dependence of β_m,k on temperature and estimate the temperature corresponding to the critical parameter value (β^*_m,k).

Figure. Representative state variable (x_i ) trajectories and bifurcation diagram for the full parameter (β_m,k ) domain.

REFERENCES

Dhooge, A., Govaerts, W., Kuznetsov, Y.A., MATCONT: A MATLAB package for numerical bifurcation analysis of ODEs. ACM T. Math. Software 29(2): 141-164 (2003).

Gerogiorgis, D.I., Dynamics and bifurcation analysis of the heat-shock response in eukaryotic cells, Proceedings of FOSBE 2009 (ed.: M. Henson), in press (2009).

Kline, M.P.,
Morimoto, R.I., Repression of the Heat Shock Factor 1 transcriptional activation domain is modulated by constitutive phosphorylation, Mol. Cell. Biol. 17(4): 2107-2115 (1997).

Lindquist, S., The heat-shock response, Ann. Rev. Biochem. 55: 1151-1191 (1986).

Muchowski, P.J., Wacker, J.L., Modulation of neurodegeneration by molecular chaperones, Nat. Neurosci. 6(1): 11-22 (2005).

Morimoto, R.I., Cells in stress: Transcriptional activation of heat-shock genes, Science 259: 1409-1410 (1993).

Rieger, T.R.,
Morimoto, R.I., Hatzimanikatis, V., Mathematical modeling of the eukaryotic heat-shock response: Dynamics of the hsp-70 promoter, Biophys. J. 88(3): 1646-1658 (2005).

Seydel, R., Practical Bifurcation and Stability Analysis ? From Equilibrium to Chaos, Springer,
USA (1994).

Szilágyi, A., Kardos, J., Osváth, S. et al., Protein folding, in: Handbook of Neurochemistry and Molecular Neurobiology (A. Lajtha, N. Banik, eds), Springer,
USA (2007).