Frequency dependence of the proton magnetic relaxation in aqueous solutions iof haemoproteins

The longitudinal proton magnetic relaxation times T₁ were measured for ferri (met)-and carbonmonoxy-bovine haemoglobin and equine myoglobin in 0.1 M KH₂PO₄ aqueous solutions near pH 6 at 5°C and 35°C from 1.5- to 60-MHz Larmor frequencies. It is concluded that the correlation time τ_C for the dipole–dipole interaction of electron and nuclear spins is in fact the electron (ferric) spin relaxation time τ_S being close to 1.5 × 10^?10 sec for both metHb and metMb at 5°C. At 35°C the paramagnetic relaxation rates are not determined solely by the relaxation of protons exchanging from the haem pocket with bulk solvent. Hence, τC at 35°C cannot be calculated from the dispersion data obtained at this temperature. The relevance of this for the determination of interspin distances r is discussed.     

Quasielastic light scattering: Effect of ionic strength on the internal dynamics of DNA

Quasielastic light scattering is used to study the effect of ionic strength on the dynamic behaviour of DNA. In a first approach the spectrum of scattered light is analyzed in terms of a single relaxation process. The large difference between the observed behaviour and that expected according to a pure diffusional process reflects the contribution associated with internal modes, which increases with decreasing ionic strength. Such behaviour is better analyzed in terms of a double relaxation process by using two relaxation times, the reciprocals of which are equal to DK² and DK² + τ_i^?1 (K), respectively, where τ_i (K) is an average value describing the set of modes observed at a given K value. Relative intensity and relaxation times, which are the more accurate parameters, were used to interpret the results. The observed increase of the relative contribution of internal modes with decreasing ionic strength is actually a relative decrease of the diffusional contribution induced by a corresponding increase of the radius of gyration R_G. On the other hand, the reciprocal τ_i^?1 (K) of the relaxation time is a linear function of K² in the analyzed KR_G range and is insensitive to ionic strength between 10^?2M and 1M. These results, when discussed according to Rouse's model, lead to define for each value of τ_i^?1 (K) a corresponding mean-squared equilibrium length 〈μ〉 which is found to be a linear function of K^?2.     

Relationship between proton–proton nmr coupling constants and substituent electronegativities. III. Conformational analysis of proline rings in solution using a generalized Karplus equation

The relationship between published vicinal proton–proton coupling constants and the pseudorotation properties of the pyrrolidine ring in L -proline, 4-hydroxy-L -proline, 4-fluoro-L -proline, and several linear and cyclic model proline peptides is investigated. Compared to earlier studies, several important improvements are incorporated: (1) a new empirical generalization of the classical Karplus equation is utilized, which allows a valid correction for the effects of electronegativity and orientation of substitutents on ³J_HH; (2) an empirical correlation between proton–proton torsion angles and the pseudorotational parameters P and τ_m is derived; and (3) the best fit of the conformational parameters to the experimental coupling constants is obtained by means of a computerized iterative least-squares procedure. Two pseudorotation ranges were considered, classified as type N (χ₂ positive sign) and type S (χ₂ negative sign). The conformational equilibrium is fully described in terms of four geometrical parameters (P_N, τ_N, P_S, τ_S) and the equilibrium constant K. The present results indicate that, in general, the geometrical properties found in x-ray studies of proline and hydroxyproline residues are well preserved in solution. Several novel features are encountered, however. It is demonstrated that the proline ring occurs in a practically 1:1 conformational equilibrium between well-defined N- and S-type forms. Introduction of an amide group at the C-terminal end has no observable effect on this equilibrium, but the formation of a peptide bond at the imino nitrogen site results in a pronounced, but not exclusive, preference for an S-type form which is roughly 1.1 kcal/mol more stable than its N-type counterpart. The hydroxyproline ring system in neutral or acidic medium displays a pure N-type state, but N-acetylation results in the appearance of a minor (S-type) conformation. Cyclic proline dipeptides similarly exist in a biased conformational equilibrium. The major form (77–88%) corresponds to the N-type conformer observed in the solid state; the minor S-form has not been observed before. In contrast, cyclic hydroxyproline dipeptides display complete conformational purity. Ranges of endocyclic torsion angles deduced for the various classes of pyrrolidine derivatives in solution are presented. Each torsion appears confined to a surprisingly narrow range, comprising about 4°–8° in most cases. In all, the proline ring is far less “floppy” than hitherto assumed.     

Syn–anti equilibrium of purine bases in dinucleoside monophosphates as studied by proton longitudinal relaxation

The preferential orientations of the purine bases in dinucleoside monophosphates such as ApA, ApG, and GpA in 10^?2M neutral aqueous solutions have been investigated by proton relaxation at 250 MHz. These orientations are deduced from computer simulations of the magnetization recovery curves following a 180° nonselective pulse. The distances between the H(8) proton of a base and the ribose ring protons which are used in these calculations are obtained by minimization as a function of the glycosyl torsion angle ? of the standard deviation between the isotropic reorientation correlation times τ_R derived from the relaxation rates of these protons. The average H(1′) – H(8) distance obtained by this procedure may be readily verified from the reduction of the H(1′) relaxation rate when H(8) is substituted by a deuteron. The limits of validity of the assumption of a single correlation time τ_R governing the proton relaxation have been estimated, taking into account several possible internal motions, e.g., the rotation of the base, of the methylene exocyclic group and the N ? S interconversion of the ribose ring. For 10^?10 < τ_R < 2 × 10^?10 sec, it appears that the influence of these motions on the proton relaxation becomes perceptible when the jump rates among equilibrium positions exceed ca. 10⁹ sec^?1. The whole of the experimental results show that for the ribose ring N conformer, the orientation of the bases is found in the ranges 60° < ? < 80° (syn) and 180° < ? < 210° (anti). For ribose S conformer, it is observed that this orientation is mainly syn with 5° < ? < 90°. The average H(1′) – H(8) distance provides semiquantitative information on the overall syn or anti orientations of the base in each nucleoside moiety. At 298 K the population of the anti conformer is found to increase in the order A- pG < Ap -G ～ Gp -A < Ap -A < A-pA < G-pA . A more detailed analysis of relaxation data shows that the maximum possible fraction of the stacked form of dinucleotides, due to the occurrence of N-anti conformers in both nucleoside moieties, is in the order ApG < GpA < ApA, in agreement with previous works, with however smaller values. Lastly the deuteron linewidth in position 8 of the bases indicates a syn–anti transition rate of the order of 10⁹ sec^?1 at room temperature, without noticeable effects therefore on the proton relaxation.     

31P nmr spin-lattice relaxation studies of deoxyoligonucleotides

Temperature-dependent conformational transitions of deoxyoligonucleotides have been monitored by measuring ³¹P chemical shifts, spin-lattice relaxation times (T₁), and ³¹P-{H} nuclear Overhauser enhancements (NOEs). The measured NOE ranged from 30 to 80%, compared to the theoretical maximum of 124% for a dipolar relaxation mediated by rapid isotropic rotation. The observed 3′-5′ phosphate diester ³¹P T₁ showed a similar temperature dependence over the range 2–75°C for both double- and single-stranded oligonucleotides, and for dinucleotides. The results show that dipole–dipole interactions dominate the internucleotide phosphate relaxation rate in oligonucleotides. The same is true of terminal phosphate groups at low temperature; but at higher temperature another process, possibly due to contamination by paramagnetic ions, becomes dominant. The rotational correlation time τ_R calculated from the dipole–dipole relaxation rate of the internucleotide phosphate in d(pA)₂ at 16°C is τ_R = 5.0 × 10^?10 sec, implying a Stokes radius for isotropic rotation of 7.6 Å. The T₁ and NOE values for the double-helical octanucleotide d(pA)₃pGpC(pT)₃ are consistent with dominance of dipole–dipole relaxation and isotropic rotation of a sphere of radius 14 Å, a reasonable dimension for the double helix. Activation energies for the rotation of dinucleotides range from 4 to 6 kcal/mol, close to the value of 4 kcal/mol expected for isotropic rotation. In order to test the possible effect of internal motion of correlation time τ_G on the results, we considered a model in which the nucleotide chain rotates about the P-O bonds. Comparison of the calculation with our experimental results shows that internal motion with τ_G ? 10^?9 sec, as found from other studies to be present for large nucleic acids, would not influence out T₁ and NOE values enough to be distinguished from isotropic rotation. However, we can conclude that τ_G cannot be as fast as 10^?10 sec, even for dinucleotides.     

Dielectric relaxation of DNA solutions. IV. Effects of salts and dyes

The effects of salts (NaCl, LiCl, Me₄NCl, AgNO₃, MgCl₂, CuCl₂ and MnCl₂) and dyes (acridine orange and methylene blue) on the low-frequency dielectric relaxation (0.1 Hz–30 kHz) of dilute aqueous solutions of DNA were investigated with varying salt or dye concentrations. Both the dielectric relaxation time τ_D and the rotational relaxation time τ estimated from the reduced viscosity decrease in quite parallel ways with increasing M/P (M/P being the normality ratio of cation to phosphate residue), reflecting the contraction of DNA molecule due to electrostatic shielding and cation binding. The agreement between τ_D and τ through the whole range of M/P supports our previous conclusion that the low-frequency relaxation of DNA arises from rotation of the molecule. The dielectric increment Δε also decreases with increasing M/P on account of both the contraction of DNA and the decrease in effective degree of dissociation of DNA. Δε as a function of M/P is interpreted in terms of a quasi-permanent dipole due to counterion fluctuation. These effects of cations are the strongest for divalent cations and rather weak for Na⁺, Li⁺, and Me₄N⁺. Effects of dye on τ_D and Δε are also well explained by the rotation of DNA molecule with a quasi-permanent dipole due to counterion fluctuation on the basis of intercalation of dye at D/P < 0.2 (D/P being the molarity ratio of dye to phosphate residue) and external binding at 0.2 < D/P < 1.0.     

Structural flexibility and fast proton transfer reflected by the dielectric properties of poly-L-proline in aqueous solution

Dielectric dispersion curves for the helix-11 form of poly-L-proline in aqueous solution have been determined for various pH in the acid range of zwitterion formation. The results could be excellently described by means of a Cole-Cole dispersion function involving the three parameters Δ?⁰ (total dielectric increment), τ_r (effective rotational relaxation time) and h (characterizing the width of the dispersion region). The quantities τ_r, and h were found to be clearly independent of pH and added inert electrolyte. An analysis of the data permits an evaluation of the dipole moments and leads to the conclusion that the molecule cannot be considered to be a completely stretched rigid rod but must be more or less bent. Addition of formic acid slightly below pH 4 caused a distinct broadening of the experimental curves which could be quantitatively interpreted by a second dielectric relaxation process due to orientation of zwitterions by means of fast proton transfer.     

Direct amide <Superscript>15</Superscript>N to <Superscript>13</Superscript>C transfers for solid-state assignment experiments in deuterated proteins

The assignment of protein backbone and side-chain NMR chemical shifts is the first step towards the characterization of protein structure. The recent introduction of proton detection in combination with fast MAS has opened up novel opportunities for assignment experiments. However, typical 3D sequential-assignment experiments using proton detection under fast MAS lead to signal intensities much smaller than the theoretically expected ones due to the low transfer efficiency of some of the steps. Here, we present a selective 3D experiment for deuterated and (amide) proton back-exchanged proteins where polarization is directly transferred from backbone nitrogen to selected backbone or sidechain carbons. The proposed pulse sequence uses only ¹H–¹⁵N cross-polarization (CP) transfers, which are, for deuterated proteins, about 30% more efficient than ¹H–¹³C CP transfers, and employs a dipolar version of the INEPT experiment for N–C transfer. By avoiding H^N–C (H^N stands for amide protons) and C–C CP transfers, we could achieve higher selectivity and increased signal intensities compared to other pulse sequences containing long-range CP transfers. The REDOR transfer is designed with an additional selective π pulse, which enables the selective transfer of the polarization to the desired ¹³C spins.     

Parallel acquisition of 3D-HA(CA)NH and 3D-HACACO spectra

We present here an NMR pulse sequence with 5 independent incrementable time delays within the frame of a 3-dimensional experiment, by incorporating polarization sharing and dual receiver concepts. This has been applied to directly record 3D-HA(CA)NH and 3D-HACACO spectra of proteins simultaneously using parallel detection of ¹H and ¹³C nuclei. While both the experiments display intra-residue backbone correlations, the 3D-HA(CA)NH provides also sequential ‘i ? 1 → i’ correlation along the ¹Hα dimension. Both the spectra contain special peak patterns at glycine locations which serve as check points during the sequential assignment process. The 3D-HACACO spectrum contains, in addition, information on prolines and side chains of residues having H–C–CO network (i.e., ¹Hβ, ¹³Cβ and ¹³COγ of Asp and Asn, and ¹Hγ, ¹³Cγ and ¹³COδ of Glu and Gln), which are generally absent in most conventional proton detected experiments.     

Arginine side chain stacking with peptide plane stabilizes the protein helix conformation in a cooperative way

A combined experimental and computational study is performed for arginine side chain stacking with the protein α‐helix. Theremostability measurements of Aristaless homeodomain, a helical protein, suggest that mutating the arginine residue R106, R137 or R141, which has the guanidino side chain stacking with the peptide plane, to alanine, destabilizes the protein. The R‐PP stacking has an energy of ～0.2‐0.4 kcal/mol. This stacking interaction mainly comes from dispersion and electrostatics, based on MP2 calculations with the energy decomposition analysis. The calculations also suggest that the stacking stabilizes 2 backbone‐backbone h‐bonds (i→i‐4 and i‐3→i‐7) in a cooperative way. Desolvation and electrostatic polarization are responsible for cooperativity with the i→i‐4 and i‐3→i‐7 h‐bonds, respectively. This cooperativity is supported by a protein α‐helices h‐bond survey in the pdb databank where stacking shortens the corresponding h‐bond distances.     

EVOLUTION OF THE MAGNITUDE AND TIMING OF INBREEDING DEPRESSION IN PLANTS

Estimates of inbreeding depression obtained from the literature were used to evaluate the association between inbreeding depression and the degree of self-fertilization in natural plant populations. Theoretical models predict that the magnitude of inbreeding depression will decrease with inbreeding as deleterious recessive alleles are expressed and purged through selection. If selection acts differentially among life history stages and deleterious effects are uncorrelated among stages, then the timing of inbreeding depression may also evolve with inbreeding. Estimates of cumulative inbreeding depression and stage-specific inbreeding depression (four stages: seed production of parent, germination, juvenile survival, and growth/reproduction) were compiled for 79 populations (using means of replicates, N = 62) comprising 54 species from 23 families of vascular plants. Where available, data on the mating system also were collected and used as a measure of inbreeding history. A significant negative correlation was found between cumulative inbreeding depression and the primary selfing rate for the combined sample of angiosperms (N = 35) and gymnosperms (N = 9); the correlation was significant for angiosperms but not gymnosperms examined separately. The average inbreeding depression in predominantly selfing species (δ = 0.23) was significantly less (43%) than that in predominantly outcrossing species (δ = 0.53). These results support the theoretical prediction that selfing reduces the magnitude of inbreeding depression. Most self-fertilizing species expressed the majority of their inbreeding depression late in the life cycle, at the stage of growth/reproduction (14 of 18 species), whereas outcrossing species expressed much of their inbreeding depression either early, at seed production (17 of 40 species), or late (19 species). For species with four life stages examined, selfing and outcrossing species differed in the magnitude of inbreeding depression at the stage of seed production (selfing δ = 0.05, N = 11; outcrossing δ = 0.32, N = 31), germination (selfing δ = 0.02, outcrossing δ = 0.12), and survival to reproduction (selfing δ = 0.04, outcrossing δ = 0.15), but not at growth and reproduction (selfing δ = 0.21, outcrossing δ = 0.27); inbreeding depression in selfers relative to outcrossers increased from early to late life stages. These results support the hypothesis that most early acting inbreeding depression is due to recessive lethals and can be purged through inbreeding, whereas much of the late-acting inbreeding depression is due to weakly deleterious mutations and is very difficult to purge, even under extreme inbreeding.     

The functional responses of adaptive consumers of two resources

This article investigates some aspects of the shape of the functional responses of consumers that utilize two resources. Adaptive variation in consumption behavior is shown to have a major effect on the relationship between amount of resource available and its rate of consumption by an average consumer individual. The effects of adaptive variation are dependent on the nutritional status of the two resources. If the resources are linearly substitutable, increases in the density of resource i will usually increase the quantity, functional response on i divided by density of i, and increases in the density of resource j will decrease this quantity. The result is that the functional response to resource i will generally decrease with the density of resource j, and will increase faster than it would otherwise have increased with the density of resource i. If resources are nonsubstitutable, an adaptive functional response to resource i will increase with the density of resource j, and it will increase more slowly with the density of resource i than it would have without adaptive change. If resources are both complementary and substitutable, the functional response will exhibit ranges of smooth change separated by rapid jumps between values, and different ranges of resource densities will result in a functional response with the characteristics of linearly substitutable or of non-substitutable resources. Adaptive functional response shape is dependent upon the tradeoff involved in raising each functional response. These results have implications for the types of indirect interactions that occur between resources as the result of a common consumer's functional response. They also suggest that the adaptive response of competing consumers to each other will differ depending on the nutritional status of the resources for which they are competing. Implications of these findings for consumer growth isocline shape and several other issues are explored.     

Charge polarization imposed by the binding site facilitates enzymatic redox reactions of quinone

Quinone reduction site (Q_i) of cytochrome bc₁ represents one of the canonical sites used to explore the enzymatic redox reactions involving semiquinone (SQ) states. However, the mechanism by which Q_i allows the completion of quinone reduction during the sequential transfers of two electrons from the adjacent heme b_H and two protons to C1- and C4-carbonyl remains unclear. Here we established that the SQ coupled to an oxidized heme b_H is a dominant intermediate of catalytic forward reaction and, contrary to the long-standing assumption, represents a significant population of SQ detected across pH 5–9. The pH dependence of its redox midpoint potential implicated proton exchange with histidine. Complementary quantum mechanical calculations revealed that the SQ anion formed after the first electron transfer undergoes charge and spin polarization imposed by the electrostatic field generated by histidine and the aspartate/lysine pair interacting with the C4- and C1-carbonyl, respectively. This favors a barrierless proton exchange between histidine and the C4-carbonyl, which continues until the second electron reaches the SQ_i. Inversion of charge polarization facilitates the uptake of the second proton by the C1-carbonyl. Based on these findings we developed a comprehensive scheme for electron and proton transfers at Q_i featuring the equilibration between the anionic and neutral states of SQ_i as means for a leak-proof stabilization of the radical intermediate. The key catalytic role of the initial charge/spin polarization of the SQ anion at the active site, inherent to the proposed mechanism, may also be applicable to the other quinone oxidoreductases.     

Polarization of counterions in polyelectrolytes

A theory of the polarization of counterions bound to a polyion, such as a DNA, in low and high electric field strengths is developed using statistical mechanics of inhomogeneous systems. For low fields, one finds that the polarizability p is (Zq)² ρ0βL³/(12[1 + Lρ0σ(L, b, ζ, Z, I, ρ0)]J), where σ = ∫¹₀ (λ′ − λ₀ {dc(λ − λ′)/dλ}_{λ = λ0} dλ′J), Z and L are the valence and the length of the polyion, respectively, q is the proton charge, β = 1/k_BT, T is the temperature, k_B is the Boltzmann constant, I is the ionic strength, λ = x/L and λ₀ = x₀/L are scaled distances, x₀ is a reference point such that the inhomogeneous counterion density at x₀ is equal to ρ₀—the uniform density in the absence of an electric field E—and c(x) is the direct correlation function of the homogeneous counterion-polyion phase, which includes attractive and repulsive interactions. If Lσ(L, .) is much less than one, then the polarizability is proportional to L³. If the term Lσ(L, .) is much larger than one, the polarizability scales as L². The induced dipole moment saturates and its value is the same as that of Mandel-Manning theories. The onset of the saturation, however, depends critically on the direct correlation function and hence polyelectrolyte effects. In the formalism, the polarization of the counterions is the equilibrium response to an electric field provided E is less than E_saturated. A dynamical scheme that incorporates the fact that in high fields the bound counterions conduct is discussed. © 1996 John Wiley & Sons, Inc.     

Frequency dependence of water proton longitudinal nuclear magnetic relaxation times in mouse tissues at 20°C

We have measured the proton longitudinal relaxation times of tissue water of healthy and tumor-bearing mice as a function of the Larmor frequency in the range 6.7 to 90 MHz. These data can be rationalized according to , where A and B are constants specific to the tissue species. We present an interpretation of this frequency dependence within the Fast Exchange Two States model. It is shown that involving a distribution of correlation times for water proton-proton interaction does not yield consistent results, whereas a physically meaningful translational diffusion model pertinent to the dipolar interaction between water protons and macromolecules protons leads to the required frequency dependence. Essentially tissues would differ by the ‘bound’ versus ‘free’ proportion, or by structural properties of cells, rather than by the time-scales governing water motion.     

The volumes of some compartment systems with sampling and loss from one compartment

If the concentrationc ₁(t)=∑ _i=1 ⁿ A _i exp (−α _i t) for one compartment, one presumes a linear catenaryn-compartment system without sinks and loss only from the same compartment, then the volumesV _i , rate constantsk _ij , and concentrationsc _i (t) in each compartment can be determined in terms of theA _i 's,A _i ′s, α _i ′s, the dose injectedD _o and the partition coefficientsr _ij =k _ij /k _ji . If the concentration would become uniform at equilibrium, then the total volume of distribution may be determined without knowledge ofr _ij or restriction to catenary configuration.     

Dielectric relaxation of DNA solutions. II

Real and imaganiry parts of complex dielectric constant of dilute solutions of DNA in 10^?3M NaCl with molecular weight ranging from 0.4 × 10⁶ to 4 × 10⁶ were measured at frequencies from 0.2 Hz to 30 kHz. Dielectric increments Δε were obtained from Cole-Cole plots and relaxation times τ_D from the loss maximum frequency. The τ_D of all samples agrees well with twice of the maximum viscoelastic relexation time in the Zimm theory, indicating that the low-frequency dielectric relaxiation should be ascribed to be the rotation of DNA. The rms dipole moment, which was obtained from Δε, agree well with that calculated from the counterion fluctuation theory. The dielectric increment was found to be greatly depressed in MgCl₂, which is resonably interpreted in terms of a strong binding of Mg⁺⁺ ions with DNA.     

On the analogy of stability of tangential discontinuities in hydrodynamics and in nonisothermal collisionless plasma

Analogy between stability of tangential discontinuities in an ideal liquid in conventional hydrodynamics and in collisionless nonisothermal plasma with T _e ≫ T _i is shown. The difference is due to specific features of dispersion of ion sound waves in nonisothermal plasma in the short-wavelength range, in which the lasma quasineutrality is violated and the dispersion curve tends to the ion Langmuir frequency.     

Conformational analysis of the type II and type III collagen α-1 chain N-telopeptides by 1H-NMR spectroscopy and restrained molecular mechanics calculations

The type II and type III collagen α-1 chain N-telopeptides are a nonadecamer with the sequence pEMAGGFDEKAGGAQLGVMQ-NH₂ and a tetradecamer with the sequence pEYEAYDVKSGVAGG-NH₂, respectively. Their conformations have been studied in CD₃OH/H₂O (60/40) solution by means of two-dimensional proton nmr spectroscopy. Based on double quantum filtered correlation spectroscopy, total correlation spectroscopy, rotating frame nuclear Overhauser enhancement (ROE) spectroscopy, and nuclear Over-hauser enhancement (NOE) spectroscopy experiments, all resonances were assigned and the conformational properties were analyzed in terms of vicinal NH-H_α coupling constants, sequential and medium-range NOEs (ROEs), and amide proton temperature coefficients. The NOE distance constraints as well as dihedral constraints based on the vicinal NH-H_α coupling constants were used as input parameters for restrained molecular mechanics, consisting of restrained molecular dynamics and restrained energy minimization calculations. The type II N-telopeptide's conformation is dominated by a fused βγ-turn between Phe⁶ and Ala¹⁰, stabilized by three hydrogen bonds and a salt bridge between the side-chain end groups of Glu⁸ and Lys⁹. The first 5 amino acids are extended with a much higher degree of conformational freedom. The 2 Gly residues following the turns were found to be highly flexible (hinge-like), leaving the spatial position of the second half of the molecule relative to the fused βγ-turn undefined. In the type III telopeptide, a series of sequential NH(i)-NH(i + 1) ROEs were observed between the amino acids Tyr² and Ser⁹, indicating that a fraction of the conformational space is helical. However, the absence of medium-range ROEs and the lack of regularity of the effects associated with α-helices suggest the presence of a nascent rather than a complete helix. © 1993 John Wiley & Sons, Inc.     

Suppression of the statistical coil state during the α ↔ β transition in homopolypeptides

A matrix formulation of the conformational partition function has been used to examine helix ? sheet transitions in homopolyamino acids. α-Helices are weighted by Zimm-Bragg parameters σ and s. Antiparallel β-sheets with tight bends are weighted by the parameters t, δ, and τ, where t is the propagation parameter. In addition, each bend contributes a factor δ, and each residue in the sheet that does not have a partner in the preceding strand contributes a factor τ. The helix can be the dominant conformation in a long chain only if two conditions are satisfied simultaneously: (i) s > 1 , and (ii) either s > t, or σ, δ, and τ are assigned values that inflict a greater penalty on antiparallel sheets than on helices. The maximum amount of coil developed during the helix ? sheet transition is strongly influenced by the size of τ, but it is only weakly dependent on the size of δ. Previously reported optical rotatory dispersion, CD, laser Raman, and nmr studies of thermally induced α ? β transitions in homopolyamino acids, notably poly(L -lysine), demonstrate that little random coil is present. If the random coil content is to remain small during the helix ? sheet transition, τ must be significantly less than unity. A small value for τ means that there is a significant penalty assessed to lysyl residues in an antiparallel sheet that do not have a partner in a preceding strand.