Shape anisotropy of a single random-walk polymer

Random walks have been used to describe a wide variety of systems ranging from cell colonies to polymers. Sixty-five years ago, Kuhn [Kuhn, W. (1934) Kolloid-Z. 68, 2–11] made the prediction, backed later by computer simulations, that the overall shape of a random-walk polymer is aspherical, yet no experimental work has directly tested Kuhn's general idea and subsequent computer simulations. By using fluorescence microscopy, we monitored the conformation of individual, long, random-walk polymers (fluorescently labeled DNA molecules) at equilibrium. We found that a polymer most frequently adopts highly extended, nonfractal structures with a strongly anisotropic shape. The ensemble-average ratio of the lengths of the long and short axes of the best-fit ellipse of the polymer was much larger than unity.

The kinesin walk: a dynamic model with elastically coupled heads.

Recently individual two-headed kinesin molecules have been studied in in vitro motility assays revealing a number of their peculiar transport properties. In this paper we propose a simple and robust model for the kinesin stepping process with elastically coupled Brownian heads that show all of these properties. The analytic and numerical treatment of our model results in a very good fit to the experimental data and practically has no free parameters. Changing the values of the parameters in the restricted range allowed by the related experimental estimates has almost no effect on the shape of the curves and results mainly in a variation of the zero load velocity that can be directly fitted to the measured data. In addition, the model is consistent with the measured pathway of the kinesin ATPase.

The load dependence of kinesin’s mechanical cycle

Kinesin is a dimeric motor protein that transports organelles in a stepwise manner toward the plus-end of microtubules by converting the energy of ATP hydrolysis into mechanical work. External forces can influence the behavior of kinesin, and force-velocity curves have shown that the motor will slow down and eventually stall under opposing loads of ≈5 pN. Using an in vitro motility assay in conjunction with a high-resolution optical trapping microscope, we have examined the behavior of individual kinesin molecules under two previously unexplored loading regimes: super-stall loads (>5 pN) and forward (plus-end directed) loads. Whereas some theories of kinesin function predict a reversal of directionality under high loads, we found that kinesin does not walk backwards under loads of up to 13 pN, probably because of an irreversible transition in the mechanical cycle. We also found that this cycle can be significantly accelerated by forward loads under a wide range of ATP concentrations. Finally, we noted an increase in kinesin’s rate of dissociation from the microtubule with increasing load, which is consistent with a load dependent partitioning between two recently described kinetic pathways: a coordinated-head pathway (which leads to stepping) and an independent-head pathway (which is static).

Charge transfer and transport in DNA

We explore charge migration in DNA, advancing two distinct mechanisms of charge separation in a donor (d)–bridge ({Bj})–acceptor (a) system, where {Bj} = B1,B2, … , BN are the N-specific adjacent bases of B-DNA: (i) two-center unistep superexchange induced charge transfer, d*{Bj}a → d∓{Bj}a±, and (ii) multistep charge transport involves charge injection from d* (or d+) to {Bj}, charge hopping within {Bj}, and charge trapping by a. For off-resonance coupling, mechanism i prevails with the charge separation rate and yield exhibiting an exponential dependence ∝ exp(−βR) on the d-a distance (R). Resonance coupling results in mechanism ii with the charge separation lifetime τ ∝ Nη and yield Y ≃ (1 + δ̄ Nη)−1 exhibiting a weak (algebraic) N and distance dependence. The power parameter η is determined by charge hopping random walk. Energetic control of the charge migration mechanism is exerted by the energetics of the ion pair state d∓B1±B2 … BNa relative to the electronically excited donor doorway state d*B1B2 … BNa. The realization of charge separation via superexchange or hopping is determined by the base sequence within the bridge. Our energetic–dynamic relations, in conjunction with the energetic data for d*/d− and for B/B+, determine the realization of the two distinct mechanisms in different hole donor systems, establishing the conditions for “chemistry at a distance” after charge transport in DNA. The energetic control of the charge migration mechanisms attained by the sequence specificity of the bridge is universal for large molecular-scale systems, for proteins, and for DNA.

A gene spans the pseudoautosomal boundary in mice

The X and Y chromosomes of the mouse, like those of other mammals, are heteromorphic over most of their length, but at the distal ends of the chromosomes is a region of sequence identity, the pseudoautosomal region (PAR), where the chromosomes pair and recombine during male meiosis. The point at which the PAR diverges into X- and Y-specific sequences is called the pseudoautosomal boundary. We have completed a genomic walk from the X-specific Amelogenin gene to the PAR. Analysis of this region revealed that the pseudoautosomal boundary of mice is located within an intron of a transcribed gene that encodes a novel RING finger protein. The first three of the exons of the gene are located on the X chromosome whereas the 3′ exons of the gene are located on both X and Y chromosomes. This unusual arrangement may indicate that the gene is in a state of transition from pseudoautosomal to X-unique and provides evidence for a process of attrition of the pseudoautosomal region on the Y chromosome.

Kinesin’s processivity results from mechanical and chemical coordination between the ATP hydrolysis cycles of the two motor domains

Kinesin is a processive motor protein: A single molecule can walk continuously along a microtubule for several micrometers, taking hundreds of 8-nm steps without dissociating. To elucidate the biochemical and structural basis for processivity, we have engineered a heterodimeric one-headed kinesin and compared its biochemical properties to those of the wild-type two-headed molecule. Our construct retains the functionally important neck and tail domains and supports motility in high-density microtubule gliding assays, though it fails to move at the single-molecule level. We find that the ATPase rate of one-headed kinesin is 3–6 s−1 and that detachment from the microtubule occurs at a similar rate (3 s−1). This establishes that one-headed kinesin usually detaches once per ATP hydrolysis cycle. Furthermore, we identify the rate-limiting step in the one-headed hydrolysis cycle as detachment from the microtubule in the ADP⋅Pi state. Because the ATPase and detachment rates are roughly an order of magnitude lower than the corresponding rates for two-headed kinesin, the detachment of one head in the homodimer (in the ADP⋅Pi state) must be accelerated by the other head. We hypothesize that this results from internal strain generated when the second head binds. This idea accords with a hand-over-hand model for processivity in which the release of the trailing head is contingent on the binding of the forward head. These new results, together with previously published ones, allow us to propose a pathway that defines the chemical and mechanical cycle for two-headed kinesin.

Torsional directed walks, entropic elasticity, and DNA twist stiffness

DNA and other biopolymers differ from classical polymers because of their torsional stiffness. This property changes the statistical character of their conformations under tension from a classical random walk to a problem we call the “torsional directed walk.” Motivated by a recent experiment on single lambda-DNA molecules [Strick, T. R., Allemand, J.-F., Bensimon, D., Bensimon, A. & Croquette, V. (1996) Science 271, 1835–1837], we formulate the torsional directed walk problem and solve it analytically in the appropriate force regime. Our technique affords a direct physical determination of the microscopic twist stiffness C and twist-stretch coupling D relevant for DNA functionality. The theory quantitatively fits existing experimental data for relative extension as a function of overtwist over a wide range of applied force; fitting to the experimental data yields the numerical values C = 120 nm and D = 50 nm. Future experiments will refine these values. We also predict that the phenomenon of reduction of effective twist stiffness by bend fluctuations should be testable in future single-molecule experiments, and we give its analytic form.

In vitro reconstruction of the Aspergillus (= Emericella) nidulans genome

A physical map of the 31-megabase Aspergillus nidulans genome is reported, in which 94% of 5,134 cosmids are assigned to 49 contiguous segments. The physical map is the result of a two-way ordering process, in which clones and probes were ordered simultaneously on a binary DNA/DNA hybridization matrix. Compression by elimination of redundant clones resulted in a minimal map, which is a chromosome walk. Repetitive DNA is nonrandomly dispersed in the A. nidulans genome, reminiscent of heterochromatic banding patterns of higher eukaryotes. We hypothesize gene clusters may arise by horizontal transfer and spread by transposition to explain the nonrandom pattern of repeats along chromosomes.

Anterograde flow of cargo across the Golgi stack potentially mediated via bidirectional “percolating” COPI vesicles

How do secretory proteins and other cargo targeted to post-Golgi locations traverse the Golgi stack? We report immunoelectron microscopy experiments establishing that a Golgi-restricted SNARE, GOS 28, is present in the same population of COPI vesicles as anterograde cargo marked by vesicular stomatitis virus glycoprotein, but is excluded from the COPI vesicles containing retrograde-targeted cargo (marked by KDEL receptor). We also report that GOS 28 and its partnering t-SNARE heavy chain, syntaxin 5, reside together in every cisterna of the stack. Taken together, these data raise the possibility that the anterograde cargo-laden COPI vesicles, retained locally by means of tethers, are inherently capable of fusing with neighboring cisternae on either side. If so, quanta of exported proteins would transit the stack in GOS 28–COPI vesicles via a bidirectional random walk, entering at the cis face and leaving at the trans face and percolating up and down the stack in between. Percolating vesicles carrying both post-Golgi cargo and Golgi residents up and down the stack would reconcile disparate observations on Golgi transport in cells and in cell-free systems.

A modified gambler's ruin model of polyethylene chains in the amorphous region.

Polyethylene chains in the amorphous region between two crystalline lamellae M unit apart are modeled as random walks with one-step memory on a cubic lattice between two absorbing boundaries. These walks avoid the two preceding steps, though they are not true self-avoiding walks. Systems of difference equations are introduced to calculate the statistics of the restricted random walks. They yield that the fraction of loops is (2M - 2)/(2M + 1), the fraction of ties 3/(2M + 1), the average length of loops 2M - 0.5, the average length of ties 2/3M2 + 2/3M - 4/3, the average length of walks equals 3M - 3, the variance of the loop length 16/15M3 + O(M2), the variance of the tie length 28/45M4 + O(M3), and the variance of the walk length 2M3 + O(M2).

Evidence for nonrandom hydrophobicity structures in protein chains.

The question of whether proteins originate from random sequences of amino acids is addressed. A statistical analysis is performed in terms of blocked and random walk values formed by binary hydrophobic assignments of the amino acids along the protein chains. Theoretical expectations of these variables from random distributions of hydrophobicities are compared with those obtained from functional proteins. The results, which are based upon proteins in the SWISS-PROT data base, convincingly show that the amino acid sequences in proteins differ from what is expected from random sequences in a statistically significant way. By performing Fourier transforms on the random walks, one obtains additional evidence for nonrandomness of the distributions. We have also analyzed results from a synthetic model containing only two amino acid types, hydrophobic and hydrophilic. With reasonable criteria on good folding properties in terms of thermodynamical and kinetic behavior, sequences that fold well are isolated. Performing the same statistical analysis on the sequences that fold well indicates similar deviations from randomness as for the functional proteins. The deviations from randomness can be interpreted as originating from anticorrelations in terms of an Ising spin model for the hydrophobicities. Our results, which differ from some previous investigations using other methods, might have impact on how permissive with respect to sequence specificity protein folding process is-only sequences with nonrandom hydrophobicity distributions fold well. Other distributions give rise to energy landscapes with poor folding properties and hence did not survive the evolution.

Least activation path for protein folding: investigation of staphylococcal nuclease folding by stopped-flow circular dichroism.

Is the pathway of protein folding determined by the relative stability of folding intermediates, or by the relative height of the activation barriers leading to these intermediates? This is a fundamental question for resolving the Levinthal paradox, which stated that protein folding by a random search mechanism would require a time too long to be plausible. To answer this question, we have studied the guanidinium chloride (GdmCl)-induced folding/unfolding of staphylococcal nuclease [(SNase, formerly EC 3.1.4.7; now called microbial nuclease or endonuclease, EC 3.1.31.1] by stopped-flow circular dichroism (CD) and differential scanning microcalorimetry (DSC). The data show that while the equilibrium transition is a quasi-two-state process, kinetics in the 2-ms to 500-s time range are triphasic. Data support the sequential mechanism for SNase folding: U3 <--> U2 <--> U1 <--> N0, where U1, U2, and U3 are substates of the unfolded protein and N0 is the native state. Analysis of the relative population of the U1, U2, and U3 species in 2.0 M GdmCl gives delta-G values for the U3 --> U2 reaction of +0.1 kcal/mol and for the U2 --> U1 reaction of -0.49 kcal/mol. The delta-G value for the U1 --> N0 reaction is calculated to be -4.5 kcal/mol from DSC data. The activation energy, enthalpy, and entropy for each kinetic step are also determined. These results allow us to make the following four conclusions. (i) Although the U1, U2, and U3 states are nearly isoenergetic, no random walk occurs among them during the folding. The pathway of folding is unique and sequential. In other words, the relative stability of the folding intermediates does not dictate the folding pathway. Instead, the folding is a descent toward the global free-energy minimum of the native state via the least activation path in the vast energy landscape. Barrier avoidance leads the way, and barrier height limits the rate. Thus, the Levinthal paradox is not applicable to the protein-folding problem. (ii) The main folding reaction (U1 --> N0), in which the peptide chain acquires most of its free energy (via van der Waals' contacts, hydrogen bonding, and electrostatic interactions), is a highly concerted process. These energy-acquiring events take place in a single kinetic phase. (iii) U1 appears to be a compact unfolded species; the rate of conversion of U2 to U1 depends on the viscosity of solution. (iv) All four relaxation times reported here depend on GdmCl concentrations: it is likely that none involve the cis/trans isomerization of prolines. Finally, a mechanism is presented in which formation of sheet-like chain conformations and a hydrophobic condensation event precede the main-chain folding reaction.

The cutoff phenomenon in finite Markov chains.

Natural mixing processes modeled by Markov chains often show a sharp cutoff in their convergence to long-time behavior. This paper presents problems where the cutoff can be proved (card shuffling, the Ehrenfests' urn). It shows that chains with polynomial growth (drunkard's walk) do not show cutoffs. The best general understanding of such cutoffs (high multiplicity of second eigenvalues due to symmetry) is explored. Examples are given where the symmetry is broken but the cutoff phenomenon persists.