Review Article
Understanding protein folding with energy landscape theory Part II: Quantitative aspects
- Steven S. Plotkin, José N. Onuchic
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- Published online by Cambridge University Press:
- 21 January 2003, pp. 205-286
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1. Introduction 206
2. Quantifying the notions behind the energy landscape 206
2.1 Basic concepts of the Random Energy Model (REM) 206
2.2 Replica-symmetric partition functions and densities of states 209
2.3 The RHP phase diagram and avoided phase transitions 210
2.4 Basic concepts of the entropy of topologically constrained polymers 212
3. Beyond the Random Energy Model 219
3.1 The GREM and the glass transition in a finite RHP 222
4. Basics of configurational diffusion for RHPs and proteins 227
4.1 Kinetics on a correlated energy landscape 231
5. Thermodynamics and kinetics of protein folding 234
5.1 A protein Hamiltonian with cooperative interactions 234
5.2 Variance of native contact energies 235
5.3 Thermodynamics of protein folding 236
5.4 Free-energy surfaces and dynamics for a Hamiltonian with pair-wise interactions 240
5.5 The effects of cooperativity on folding 242
5.6 Transition-state drift 242
5.7 Phase diagram for a model protein 245
5.8 A non-Arrhenius folding-rate curve for proteins 246
6. Non-Markovian configurational diffusion and reaction coordinates in protein folding 247
6.1 An illustrative example 250
6.2 Non-Markovian rate theory and reaction surfaces 251
6.3 Application of non-Markovian rate theory to simulation data 257
7. Structural and energetic heterogeneity in the folding mechanism 259
7.1 The general strategy 261
7.2 An illustrative example 263
7.3 Free-energy functional 264
7.4 Dependence of the barrier height on mean loop length (contact order) and structural variance 268
7.5 Illustrations using lattice model proteins and functional theory 269
7.6 Connections of functional theory with experiments 271
8. Conclusions and future prospects 273
9. Acknowledgments 274
10. Appendices
A. Table of common symbols 275
B. GREM construction for the glass transition 276
C. Effect of a Q-dependent diffusion coefficient 279
D. A frequency-dependent Einstein relation 279
11. References 281
We have seen in Part I of this review that the energy landscape theory of protein folding is a statistical description of a protein's complex potential energy surface, where individual folding events are sampled from an ensemble of possible routes on the landscape. We found that the most likely global structure for the landscape of a protein can be described as that of a partially random heteropolymer with a rugged, yet funneled landscape towards the native structure. Here we develop some quantitative aspects of folding using tools from the statistical mechanics of disordered systems, polymers, and phase transitions in finite-sized systems. Throughout the text we will refer to concepts and equations developed in Part I of the review, and the reader is advised to at least survey its contents before proceeding here. Sections, figures or equations from Part I are often prefixed with I- [e.g. Section I-1.1, Fig. I-1, Eq. (I-1.1)].
Biophysical basis of brain activity: implications for neuroimaging
- Robert G. Shulman, Fahmeed Hyder, Douglas L. Rothman
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- Published online by Cambridge University Press:
- 21 January 2003, pp. 287-325
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1. Summary 288
2. Introduction 288
3. Relationship between neuroenergetics and neurotransmitter flux 294
4. A model of coupling between neuroenergetics and neurotransmission 296
5. Relationship between neuroenergetics and neural spiking frequency 297
6. Comparison with previous electrophysiological and fMRI measurements 298
7. Contributions of non-oxidative energetics to a primarily oxidative brain 299
8. Possible explanation for non-oxidative energetics contributions 300
9. A model of total neuronal activity to support cerebral function 302
10. Implications for interpretation of fMRI studies 305
11. The restless brain 306
12. Acknowledgements 310
13. Appendix A. CMRO2by13C-MRS 310
14. Appendix B.Vcycand test of model 313
15. Appendix C. CMRO2by calibrated BOLD 316
16. Appendix D. Comparison of spiking activity of a neuronal ensemble with CMRO2318
17. References 320
In vivo13C magnetic resonance spectroscopy (MRS) studies of the brain have quantitatively assessed rates of glutamate–glutamine cycle (Vcyc) and glucose oxidation (CMRGlc(ox)) by detecting 13C label turnover from glucose to glutamate and glutamine. Contrary to expectations from in vitro and ex vivo studies, the in vivo13C-MRS results demonstrate that glutamate recycling is a major metabolic pathway, inseparable from its actions of neurotransmission. Furthermore, both in the awake human and in the anesthetized rat brain, Vcyc and CMRGlc(ox) are stoichiometrically related, where more than two thirds of the energy from glucose oxidation supports events associated with glutamate neurotransmission. The high energy consumption of the brain measured at rest and its quantitative relation to neurotransmission reflects a sizeable activity level for the resting brain. The high activity of the non-stimulated brain, as measured by cerebral metabolic rate of oxygen use (CMRO2), establishes a new neurophysiological basis of cerebral function that leads to reinterpreting functional imaging data because the large baseline signal is commonly discarded in cognitive neuroscience paradigms. Changes in energy consumption (ΔCMRO2%) can also be obtained from magnetic resonance imaging (MRI) experiments, using the blood oxygen level- dependent (BOLD) image contrast, provided that all the separate parameters contributing to the functional MRI (fMRI) signal are measured. The BOLD-derived ΔCMRO2% when compared with alterations in neuronal spiking rate (Δν%) during sensory stimulation in the rat reveals a stoichiometric relationship, in good agreement with 13C-MRS results. Hence fMRI when calibrated so as to provide ΔCMRO2% can provide high spatial resolution evaluation of neuronal activity. Our studies of quantitative measurements of changes in neuroenergetics and neurotransmission reveal that a stimulus does not provoke an arbitrary amount of activity in a localized region, rather a total level of activity is required where the increment is inversely related to the level of activity in the non-stimulated condition. These biophysical experiments have established relationships between energy consumption and neuronal activity that provide novel insights into the nature of brain function and the interpretation of fMRI data.