Theory on the rate equations of Michaelis-Menten type enzyme kinetics with competitive inhibition.

We derive approximate expressions for pre- and post-steady state regimes of the velocity-substrate-inhibitor spaces of the Michaelis-Menten enzyme kinetic scheme with fully and partial competitive inhibition. Our refinement over the currently available standard quasi steady state approximation (sQSSA) seems to be valid over wide range of enzyme to substrate and enzyme to inhibitor concentration ratios. Further, we show that the enzyme-inhibitor-substrate system can exhibit temporally well-separated two different steady states with respect to both enzyme-substrate and enzyme-inhibitor complexes under certain conditions. We define the ratios fS = vmax/(KMS + e0) and fI = umax/(KMI + e0) as the acceleration factors with respect to the catalytic conversion of substrate and inhibitor into their respective products. Here KMS and KMI are the Michaelis-Menten parameters associated respectively with the binding of substrate and inhibitor with the enzyme, vmax and umax are the respective maximum reaction velocities and e0, s0, and i0 are total enzyme, substrate and inhibitor levels. When (fS/fI) < 1, then enzyme-substrate complex will show multiple steady states and it reaches the full-fledged steady state only after the depletion of enzyme-inhibitor complex. When (fS/fI) > 1, then the enzyme-inhibitor complex will show multiple steady states and it reaches the full-fledged steady state only after the depletion of enzyme-substrate complex. This multi steady-state behavior especially when (fS/fI) ≠ 1 is the root cause of large amount of error in the estimation of various kinetic parameters of fully and partial competitive inhibition schemes using sQSSA. Remarkably, we show that our refined expressions for the reaction velocities over enzyme-substrate-inhibitor space can control this error more significantly than the currently available sQSSA expressions.

234U/238U disequilibrium and 235U/238U ratios measured using MC-ICP-MS in natural high background radiation area soils to understand the fate of uranium.

The Chhatrapur-Gopalpur coastal area in Odisha, India is a well-known natural high background radiation (HBRA) area due to the abundance of monazite (a thorium bearing radioactive mineral) in beach sands and soils. Recent studies on Chhatrapur-Gopalpur HBRA groundwater have reported high concentrations of uranium and its decay products. Therefore, the soils of the Chhatrapur-Gopalpur HBRA are reasonably suspected as the sources of these high uranium concentrations in groundwater. In this report, first the uranium concentrations in soil samples were measured using inductively coupled plasma mass spectrometry (ICP-MS) and they were found to range from 0.61 ± 0.01 to 38.59 ± 0.16 mg kg-１. Next, the 234U/238U and 235U/238U isotope ratios were measured to establish a baseline for the first time in Chhatrapur-Gopalpur HBRA soil. Multi-collector inductively coupled plasma mass spectrometry (MC-ICP-MS) was used for measurement of these isotope ratios. The 235U/238U ratio was observed to be the normal terrestrial value. The 234U/238U activity ratio, was calculated to understand the secular equilibrium between 234U and 238U in soil and it varied from 0.959 to 1.070. To understand the dynamics of uranium in HBRA soil, physico-chemical characteristics of soil were correlated with uranium isotope ratios and this correlation of 234U/238U activity ratio indicated the leaching of 234U from Odisha HBRA soil.

Multiple transcription auto regulatory loops can act as robust oscillators and decision-making motifs.

Response time decides how fast a gene can react against an external signal at the transcription level in a signalling cascade. The steady state protein levels of the responding genes decide the coupling between two consecutive members of a signalling cascade. A negative autoregulatory loop (NARL) present in a transcription factor network can speed up the response time of the regulated gene at the cost of reduced steady state protein level. We present here a multi NARL motif which can be tuned for both the steady state protein level as well as response time in the required direction. Remarkably, there exists an optimum Hill coefficient nop t ≅ 4 at which the response time of the NARL motif is at minimum. When the Hill coefficient is n < nopt , then under strong binding conditions, one can raise the steady state protein level by increasing the gene copy number with almost no change in the response time of the multi NARL motif. Using detailed computational analysis, we show that the coupled multi NARL and positive auto regulatory loop (PARL) motifs can act as an oscillator as well as decision making component which are robust against extrinsic fluctuations in the control parameters. We further demonstrate that the period of oscillation of the coupled multi NARL-PARL dual feedback oscillator can also be fine-tuned by the gene copy number apart from the inducer concentration. We finally demonstrate robustness of bistable dual feedback decision making motifs with multi autoregulatory loop component.

ACTIVITY RATIO OF URANIUM IN FUKUSHIMA SOIL SAMPLES USING MULTI-COLLECTOR INDUCTIVELY COUPLED PLASMA MASS SPECTROMETRY.

The Fukushima soils have been collected from Namie and Futaba areas for the radiocaesium and uranium isotope ratio studies. The 137Cs activity concentration of soil samples ranged from 6 ± 1 to 756 ± 14 kBq/kg. The uranium isotope ratios are measured using multi collector inductively coupled plasma mass spectrometry. The activity ratio (234U/238U) of the Fukushima soils is calculated from the measured 234U/238U isotope ratio. Activity ratio varied from 0.98 to 1.02 which indicates that 234U and 238U are in secular equilibrium. The 235U/238U atomic ratio of the Fukushima soils did not show any heterogeneity compared with the natural terrestrial ratio even with high level of 137Cs in soils.

ICP-MS Measurement of Trace and Rare Earth Elements in Beach Placer-Deposit Soils of Odisha, East Coast of India, to Estimate Natural Enhancement of Elements in the Environment.

Inductively coupled plasma mass spectrometry (ICP-MS) has been used to measure the concentration of trace and rare earth elements (REEs) in soils. Geochemical certified reference materials such as JLk-1, JB-1, and JB-3 were used for the validation of the analytical method. The measured values were in good agreement with the certified values for all the elements and were within 10% analytical error. Beach placer deposits of soils mainly from Odisha, on the east coast of India, have been selected to study selected trace and rare earth elements (REEs), to estimate enrichment factor (EF) and geoaccumulation index (Igeo) in the natural environment. Enrichment factor (EF) and geoaccumulation index (Igeo) results showed that Cr, Mn, Fe, Co, Zn, Y, Zr, Cd and U were significantly enriched, and Th was extremely enriched. The total content of REEs (Æ©REEs) ranged from 101.3 to 12,911.3 µg g-1, with an average 2431.1 µg g-1 which was higher than the average crustal value of ΣREEs. A high concentration of Th and light REEs were strongly correlated, which confirmed soil enrichment with monazite minerals. High ratios of light REEs (LREEs)/heavy REEs (HREEs) with a strong negative Eu anomaly revealed a felsic origin. The comparison of the chondrite normalized REE patterns of soil with hinterland rocks such as granite, charnockite, khondalite and migmatite suggested that enhancement of trace and REEs are of natural origin.

A Microwave Digestion Technique for the Analysis of Rare Earth Elements, Thorium and Uranium in Geochemical Certified Reference Materials and Soils by Inductively Coupled Plasma Mass Spectrometry.

Two different digestion methods-microwave digestion (Mw) and Savillex digestion (Sx)-were used to evaluate the best quality control for analysis of the rare earth elements, Th and U in the geochemical certified reference material JSd-2, supplied by the Geological Survey of Japan (GSJ). The analysis of trace elements was carried out using inductively coupled plasma mass spectrometry (ICP-MS). The digestion recovery was > 90% for almost all elements by both methods. Mw-4 (four repeats of the microwave digestion) was found to be more effective and faster than Sx. In order to evaluate the efficiency of Mw-4, three other GSJ certified reference materials, JLk-1, JB-1 and JB-3, as well as five different soil samples from Belarus, Japan, Serbia and Ukraine were also analyzed. The Mw-4 method was seen to be promising for complete digestion and recovery of most of the elements. The U/Th ratio showed some heterogeneity for Ukraine and Serbia soils affected by Chernobyl nuclear power plant accident and depleted uranium contamination, respectively. This method can be successfully applied to any type of soils for elemental analyses.

Chemical Separation of Uranium and Precise Measurement of 234U/238U and 235U/238U Ratios in Soil Samples Using Multi Collector Inductively Coupled Plasma Mass Spectrometry.

A new chemical separation has been developed to isolate uranium (U) using two UTEVA columns to minimize iron and thorium interferences from high background area soil samples containing minerals like monazites and ilmenite. The separation method was successfully verified in some certified reference materials (CRMs), for example, JSd-2, JLk-1, JB-1 and JB-3. The same method was applied for purification of U in Fukushima soil samples affected by the Fukushima dai-ichi nuclear power station (FDNPS) accident. Precise and accurate measurement of 234U/238U and 235U/238U isotope ratios in chemically separated U were carried out using a multi-collector inductively coupled plasma mass spectrometer (MC-ICP-MS). In this mass spectrometric method, an array of two Faraday cups (1011 Ω, 1012 Ω resistor) and a Daly detector were simultaneously employed. The precision of U isotope ratios in an in-house standard was evaluated by replicate measurement. Relative standard deviation (RSD) of 234U/238U and 235U/238U were found to be 0.094% (2σ) and 0.590% (2σ), respectively. This method has been validated using a standard reference material SRM 4350B, sediment sample. The replicate measurements of 234U/238U in SRM shows 0.7% (RSD). This developed method is suitable for separation of U and its isotope ratio measurement in environmental samples.

Precise Measurement of Tellurium Isotope Ratios in Terrestrial Standards Using a Multiple Collector Inductively Coupled Plasma Mass Spectrometry.

Precise tellurium (Te) isotope ratio measurement using mass spectrometry is a challenging task for many decades. In this paper, Te isotope ratio measurements using multi-collector inductively coupled plasma mass spectrometry (MC-ICP-MS) in terrestrial Te standards have been reported. Newly developed Faraday cup with 1012 Ω resistor is used to measure low abundance 120Te, whereas the 1011 Ω resistor is used to measure other Te isotopes. The relative standard deviation obtained for Te isotope ratio measurement by Faraday cups of 120Te/128Te [0.002907(05)], 122Te/128Te [0.079646(10)], 123Te/128Te [0.027850(07)], 125Te/128Te [0.221988(09)], 126Te/128Te [0.592202(20)], and 130Te/128Te [1.076277(30)] were 0.140%, 0.014%, 0.026%, 0.005%, 0.004%, and 0.004%, respectively. The measured isotope ratio results are compared with previous results obtained by thermal ionization mass spectrometry (TIMS), negative thermal ionization mass spectrometry (N-TIMS), and MC-ICP-MS, showing an improvement in the precision about one order of magnitude for 120Te/128Te ratio. The present study shows better precision for Te isotope ratios compared to earlier studies.

Theory of Site-Specific DNA-Protein Interactions in the Presence of Nucleosome Roadblocks.

We show that nucleosomes exert a maximal amount of hindrance to the one-dimensional diffusion of transcription factors (TFs) when they are present between TFs and their cognate sites on DNA. The effective one-dimensional diffusion coefficient of TFs (χTF) decreases with a rise in the free-energy barrier (µNU) of the sliding of nucleosomes as χTF∝exp(-µNU). The average time (ηL) required by TFs to slide over L sites on DNA increases with µNU as ηL∝exp(µNU). When TFs move close to nucleosomes, then they exhibit typical subdiffusion. Nucleosomes can enhance the search dynamics of TFs when TFs are present between nucleosomes and TF binding sites. These results suggest that nucleosome-depleted regions around the cognate sites of TFs are mandatory for efficient site-specific binding of TFs. Remarkably, the genome-wide in vivo positioning pattern of TFs shows a maximum at their specific binding sites where the occupancy of nucleosomes shows a minimum. This could be a consequence of an increasing level of breathing dynamics of nucleosome cores and decreasing levels of fluctuations in the DNA binding domains of TFs as they move across TF binding sites. The dynamics of TFs becomes slow as they approach their cognate sites so that TFs form a tight site-specific complex, whereas the dynamics of nucleosomes becomes rapid so that they quickly pass through the cognate sites of TFs. Several in vivo data sets on the genome-wide positioning pattern of nucleosomes and TFs agree well with our arguments. The retarding effects of nucleosomes can be minimized when the degree of condensation of DNA is such that it can permit a jump size associated with the dynamics of TFs beyond âˆ¼160-180 bp.

Theory on the Mechanism of DNA Renaturation: Stochastic Nucleation and Zipping.

Renaturation of the complementary single strands of DNA is one of the important processes that requires better understanding in the view of molecular biology and biological physics. Here we develop a stochastic dynamical model on the DNA renaturation. According to our model there are at least three steps in the renaturation process viz. nonspecific-contact formation, correct-contact formation and nucleation, and zipping. Most of the earlier two-state models combined nucleation with nonspecific-contact formation step. In our model we suggest that it is considerably meaningful when we combine the nucleation with the zipping since nucleation is the initial step of zipping and nucleated and zipping molecules are indistinguishable. Nonspecific contact formation step is a pure three-dimensional diffusion controlled collision process. Whereas nucleation involves several rounds of one-dimensional slithering and internal displacement dynamics of one single strand of DNA on the other complementary strand in the process of searching for the correct-contact and then initiate nucleation. Upon nucleation, the stochastic zipping follows to generate a fully renatured double stranded DNA. It seems that the square-root dependency of the overall renaturation rate constant on the length of reacting single strands originates mainly from the geometric constraints in the diffusion controlled nonspecific-contact formation step. Further the inverse scaling of the renaturation rate on the viscosity of reaction medium also originates from nonspecific contact formation step. On the other hand the inverse scaling of the renaturation rate with the sequence complexity originates from the stochastic zipping which involves several rounds of crossing over the free-energy barrier at microscopic levels. When the sequence of renaturing single strands of DNA is repetitive with less complexity then the cooperative effects will not be noticeable since the parallel zipping will be a dominant enhancing factor. However for DNA strands with high sequence complexity and length one needs to consider the underlying cooperative effects both at microscopic and macroscopic levels to explain various scaling behaviours of the overall renaturation rate.

Theory on the dynamics of oscillatory loops in the transcription factor networks.

We develop a detailed theoretical framework for various types of transcription factor gene oscillators. We further demonstrate that one can build genetic-oscillators which are tunable and robust against perturbations in the critical control parameters by coupling two or more independent Goodwin-Griffith oscillators through either -OR- or -AND- type logic. Most of the coupled oscillators constructed in the literature so far seem to be of -OR- type. When there are transient perturbations in one of the -OR- type coupled-oscillators, then the overall period of the system remains constant (period-buffering) whereas in case of -AND- type coupling the overall period of the system moves towards the perturbed oscillator. Though there is a period-buffering, the amplitudes of oscillators coupled through -OR- type logic are more sensitive to perturbations in the parameters associated with the promoter state dynamics than -AND- type. Further analysis shows that the period of -AND- type coupled dual-feedback oscillators can be tuned without conceding on the amplitudes. Using these results we derive the basic design principles governing the robust and tunable synthetic gene oscillators without compromising on their amplitudes.

Theory on the coupled stochastic dynamics of transcription and splice-site recognition.

Eukaryotic genes are typically split into exons that need to be spliced together to form the mature mRNA. The splicing process depends on the dynamics and interactions among transcription by the RNA polymerase II complex (RNAPII) and the spliceosomal complex consisting of multiple small nuclear ribonucleo proteins (snRNPs). Here we propose a biophysically plausible initial theory of splicing that aims to explain the effects of the stochastic dynamics of snRNPs on the splicing patterns of eukaryotic genes. We consider two different ways to model the dynamics of snRNPs: pure three-dimensional diffusion and a combination of three- and one-dimensional diffusion along the emerging pre-mRNA. Our theoretical analysis shows that there exists an optimum position of the splice sites on the growing pre-mRNA at which the time required for snRNPs to find the 5' donor site is minimized. The minimization of the overall search time is achieved mainly via the increase in non-specific interactions between the snRNPs and the growing pre-mRNA. The theory further predicts that there exists an optimum transcript length that maximizes the probabilities for exons to interact with the snRNPs. We evaluate these theoretical predictions by considering human and mouse exon microarray data as well as RNAseq data from multiple different tissues. We observe that there is a broad optimum position of splice sites on the growing pre-mRNA and an optimum transcript length, which are roughly consistent with the theoretical predictions. The theoretical and experimental analyses suggest that there is a strong interaction between the dynamics of RNAPII and the stochastic nature of snRNP search for 5' donor splicing sites.

Theory on the dynamics of feedforward loops in the transcription factor networks.

Feedforward loops (FFLs) consist of three genes which code for three different transcription factors A, B and C where B regulates C and A regulates both B and C. We develop a detailed model to describe the dynamical behavior of various types of coherent and incoherent FFLs in the transcription factor networks. We consider the deterministic and stochastic dynamics of both promoter-states and synthesis and degradation of mRNAs of various genes associated with FFL motifs. Detailed analysis shows that the response times of FFLs strongly dependent on the ratios (w(h) = γ(pc)/γ(ph) where h = a, b, c corresponding to genes A, B and C) between the lifetimes of mRNAs (1/γ(mh)) of genes A, B and C and the protein of C (1/γ(pc)). Under strong binding conditions we can categorize all the possible types of FFLs into groups I, II and III based on the dependence of the response times of FFLs on w(h). Group I that includes C1 and I1 type FFLs seem to be less sensitive to the changes in w(h). The coherent C1 type seems to be more robust against changes in other system parameters. We argue that this could be one of the reasons for the abundant nature of C1 type coherent FFLs.

On the minimization of fluctuations in the response times of autoregulatory gene networks.

The temporal dynamics of the concentrations of several proteins are tightly regulated, particularly for critical nodes in biological networks such as transcription factors. An important mechanism to control transcription factor levels is through autoregulatory feedback loops where the protein can bind its own promoter. Here we use theoretical tools and computational simulations to further our understanding of transcription-factor autoregulatory loops. We show that the stochastic dynamics of feedback and mRNA synthesis can significantly influence the speed of response of autoregulatory genetic networks toward external stimuli. The fluctuations in the response-times associated with the accumulation of the transcription factor in the presence of negative or positive autoregulation can be minimized by confining the ratio of mRNA/protein lifetimes within 1:10. This predicted range of mRNA/protein lifetime agrees with ranges observed empirically in prokaryotes and eukaryotes. The theory can quantitatively and systematically account for the influence of regulatory element binding and unbinding dynamics on the transcription-factor concentration rise-times. The simulation results are robust against changes in several system parameters of the gene expression machinery.

Packaging effects on site-specific DNA-protein interactions.

We show that the rate of site-specific association of a protein molecule of interest with the DNA chain can be approximately 10(2) times higher than that of the three-dimensional diffusion-controlled collision rate limit approximately 10(8) mol(-1) s(-1) only when the protein molecule of interest searches for its specific site on the DNA chain in a reduced dimensional space with a dimensionality dr of dr<1. Upon considering the concurrent dynamics of the linear DNA chain that is embedded in a d-dimensional space along with the one-dimensional diffusion dynamics of the nonspecifically bound protein molecule on the DNA chain, we derive the generalized scaling law epsilon approximately 2(3(2-d)+3), where epsilon is the number of times by which the rate of site-specific association of the protein molecule with the DNA chain can be enhanced over the three-dimensional diffusion-controlled collision rate limit and d is the dimensionality of the reduced search space. Using the analogy between the self-intersection loop length in the theory of random walks and the ring-closure events in the theory of site specific interactions of a protein molecule with the DNA chain, we further show that the extent of packaging and volume compression of the genomic DNA inside the living cell is designed in such a way that the efficiency of the protein molecule in the process of searching for its specific site on the genomic DNA is a maximum. Our simulation results suggest that the volume compression factor theta which is the ratio between the total volume of the living cell and the volume occupied by the DNA chain along with all the other bound protein molecules should be such that theta>or=100 for an efficient site specific interaction of a protein molecule of interest with the linear DNA chain that is embedded in a three-dimensional space. Our theoretical and simulation results agree well with the E. coli cellular system.

Directional dependent dynamics of protein molecules on DNA.

We demonstrate that a protein molecule of interest undergoing the one-dimensional Brownian dynamics along DNA can exhibit a directional dependent net transport either toward or away from its target site depending on the distribution of the initial positions of the other classes of protein molecules present on the same DNA. Directionality arises as a consequence of the confinement of the search space and dynamic reflections by other protein molecules present on the same DNA chain. Energy cost for such directionality comes from the free energy spent on setting the initial positions of the other protein molecules. In the mechanism of action of cis-acting elements on the initiation of transcription, such free-energy inputs are derived from the site-specific binding affinities of the inflowing transcriptional factors toward their cis-acting elements. If the initial distribution of other protein molecules is a random one, then the protein molecule of interest exhibits a net transport away from its target site. This directionality originates from unequal natures of enhancing and retarding effects of the randomly distributed other classes of protein molecules. The protein molecule of interest overcomes the retarding effects of the other classes of protein molecules in a dynamical manner by increasing the number of dissociation-association events when it is far away from its target site and then by switching back to the sliding dynamics due to increase in the enhancing effects as it moves closer to its target site.

Action at distance along the DNA: a random jump model on enhancer action.

Enhancers are important regulatory elements associated with eukaryotic genes. Here we present a random jump model on enhancer action at distance along the DNA. We show that to initiate the enhancing-action of an enhancer, a minimum jump size k=k(omega) which is directly proportional to the size of the genome, must be possessed by the RNA polymerase (RNAP) in the process of searching for the promoter sequences. When the jump size is near to or above k(omega), our model predicts that enhancers increase the level of expression of a gene mainly by increasing the probability of the gene to get transcribed rather than by increasing the transcriptional rate. Apart from this, our model also predicts that enhancer can increase the transcriptional probability only in the presence of the memory of the first time enhancer-RNAP contact. When the jump size associated with dynamics of RNAP on the DNA is close to or above certain critical values k=k(c) approximately 2N(2/3)where N is the length of the DNA under consideration, enhancers can regulate the transcription of a gene in a position and distance independent manner and at the jump size k=k(c) the enhancing action is a maximum. Since the jump size k is directly proportional to the degree of super-coiling or close-packed nature the DNA, our model suggests that to initiate the enhancer action a minimum degree of super-coiling of DNA is necessary that corresponds to the requirement of a minimum jump size k=k(omega) which agrees well with the experimental observations.

An artificial electron donor supported catalytic cycle of Pseudomonas putida cytochrome P450cam.

Putidaredoxin (PdX), the physiological effector of cytochrome P450cam (P450cam), serves to gate electron transfer into oxy-P450cam during the catalytic cycle of the enzyme. Redox-linked structural changes in PdX are necessary for the effective P450cam turnover reaction. PdX is believed to be difficult to be replaced by an artificial electron donor in the reaction pathway of P450cam. We demonstrate that the catalytic cycle of wild-type P450cam can be supported in the presence of an artificial reductant, potassium ferrocyanide. Upon rapid mixing of ferrocyanide ion with P450cam, we observed an intermediate with spectral features characteristic of compound I. The rate constant for the formation of compound I in the presence of ferrocyanide supported reaction cycle was found to be comparable to the ones observed for H2O2 supported compound I formation in wild-type P450cam, but was much lower than those observed for classical peroxidases. The results presented in this paper form the first kinetic analysis of this intermediate for an artificial electron-driven P450cam catalytic pathway in solution.

Structure and redox properties of the haem centre in the C357M mutant of cytochrome P450cam.

The effects of site-specific mutation of the axial cysteine (C357M) to a methionine residue in cytochrome P450cam on the enzyme's coordination geometry and redox potential have been investigated. The absorption spectra of the haem centre in the C357M mutant of the enzyme showed close similarity to those of cytochrome c both in the oxidised and reduced forms. A well-defined absorption peak at 695 nm, similar to that seen in the case of cytochrome c and characteristic of methionine ligation to the ferric haem, was observed. The results indicated that the haem of C357M cytochrome P450cam is possibly axially coordinated to a methionine and a histidine, analogously to cytochrome c. The circular dichroism spectra in the visible and the far-UV regions suggested that the tertiary structure of the haem cavity in the C357M mutant cytochrome P450cam was distinctly different from that in the wild-type enzyme or in cytochrome c, although the secondary structure of the mutant remained identical to that of the wild-type cytochrome P450cam. Comparison of the natures of the CD spectra in the 400 nm and 695 nm regions of the C357M mutant of cytochrome P450cam with those of horse cytochrome c suggested (R) chirality at the sulfur atom of the iron-bound methionine residue in the mutant. The redox potential of the haem centre, estimated by redox titration of the C357M mutant, was found to be +260 mV, which is much higher than that in the wild-type enzyme and similar to the redox potential of cytochrome c. This supported the concept that axial ligation of the haem plays the major role in tuning the redox potential of the haem centre in haem proteins.

A simple formalism on dynamics of proteins on potential energy landscapes.

We present a simple formalism for the dynamics of proteins on a potential energy landscape, using connectedness of configurational domains as an order parameter. This formalism clearly shows that the energy bias required to form a unit correct contact toward the native configuration of a two-state folder, to overcome Levinthal's paradox, is E(bias) congruent with RT ln 2. This result agrees well with earlier studies and indicates that the bias is mainly due to hydrophobic interaction. Further investigations have shown that the landscape funnel could be experimentally mapped onto a two-dimensional space formed by denaturant concentration and the connectedness of configurational domains. The theoretical value of the depth-of-folding funnel in terms of denaturant concentration has been calculated for a model protein (P450cam), which agrees well with the experimental value. Using our model, it is also possible to explain the turnover nature of heat-capacity change upon unfolding of proteins and the existence of enthalpy and entropy convergence temperatures during unfolding without any strict assumptions as proposed in earlier studies.