The influence of topographic microstructures on the initial adhesion of L929 fibroblasts studied by single-cell force spectroscopy

Single-cell force spectroscopy was used to investigate the initial adhesion of L929 fibroblasts onto periodically grooved titanium microstructures (height ~6 μm, groove width 20 μm). The position-dependent local adhesion strength of the cells was correlated with their rheological behavior. Spherical cells exhibited a significantly lower Young’s modulus (<1 kPa) than that reported for spread cells, and their elastic properties can roughly be explained by the Hertz model for an elastic sphere. While in contact with the planar regions of the substrate, the cells started to adapt their shape through slight ventral flattening. The process was found to be independent of the applied contact force for values between 100 and 1,000 pN. The degree of flattening correlated with the adhesion strength during the first 60 s. Adhesion strength can be described by fast exponential kinetics as C₁ [ 1 - exp( - C₂ ·t ) ] C_{1} \left[ {1 - \exp \left( { - C_{2} \cdot t} \right)} \right] with C ₁ = 2.34 ± 0.19 nN and C ₂ = 0.09 ± 0.02 s⁻¹. A significant drop in the adhesion strength of up to 50% was found near the groove edges. The effect can be interpreted by the geometric decrease of the contact area, which indicates the inability of the fibroblasts to adapt to the shape of the substrate. Our results explain the role of the substrate’s topography in contact guidance and suggest that rheological cell properties must be considered in cell adhesion modeling.     

Toxicity assessment of common organic solvents using a biosensor based on algal photosynthetic activity measurement

The acute toxicities of common organic solvents (e.g., methanol, ethanol, isopropanol, acetone, acetonitrile, and dimethylformamide) were evaluated using a biosensor based on microalgal photosynthesis measurement. The biosensor was air-tight, with no headspace, preventing volatile organic toxicants from escaping into the environment as well as partitioning from the aqueous phase into the headspace until equilibrium was reached. Both the incubating and exposure times were set at 10 min. It was observed that only 2 h was needed to obtain complete dose-related inhibition of photosynthetic activity. The results showed that all the tested organic solvents inhibited algal photosynthesis with EC₅₀ ranging between 589 and 2,570 mM. The inhibition of these solvents was in the order: isopropanol > acetone > acetonitrile > ethanol > dimethylformamide > methanol. The quantitative structure-activity relationship (QSAR) between toxicity data and partition coefficient of the examined compounds could be modeled as follows: ${\text{log}}_{{10}} {\text{EC}}_{{50}} \;{\left( {\mu {\text{M}}} \right)} = - 0.6428\;{\text{log}}\;P + 5.76\;{\left( {{\text{R}}^{2} \approx 0.88} \right)}The acute toxicities of common organic solvents (e.g., methanol, ethanol, isopropanol, acetone, acetonitrile, and dimethylformamide) were evaluated using a biosensor based on microalgal photosynthesis measurement. The biosensor was air-tight, with no headspace, preventing volatile organic toxicants from escaping into the environment as well as partitioning from the aqueous phase into the headspace until equilibrium was reached. Both the incubating and exposure times were set at 10 min. It was observed that only 2 h was needed to obtain complete dose-related inhibition of photosynthetic activity. The results showed that all the tested organic solvents inhibited algal photosynthesis with EC₅₀ ranging between 589 and 2,570 mM. The inhibition of these solvents was in the order: isopropanol > acetone > acetonitrile > ethanol > dimethylformamide > methanol. The quantitative structure-activity relationship (QSAR) between toxicity data and partition coefficient of the examined compounds could be modeled as follows: \textlog₁₀ \textEC₅₀   ( m\textM ) = - 0.6428  \textlog  P + 5.76  ( \textR² ? 0.88 ){\text{log}}_{{10}} {\text{EC}}_{{50}} \;{\left( {\mu {\text{M}}} \right)} = - 0.6428\;{\text{log}}\;P + 5.76\;{\left( {{\text{R}}^{2} \approx 0.88} \right)}. This indicates that the photosynthetic activity of the microalga Pseudokirchneriella subcapitata is highly dependent on the hydrophobicity of these commonly used organic solvents.     

Stability of differential susceptibility and infectivity epidemic models

We introduce classes of differential susceptibility and infectivity epidemic models. These models address the problem of flows between the different susceptible, infectious and infected compartments and differential death rates as well. We prove the global stability of the disease free equilibrium when the basic reproduction ratio R₀ ￡ 1{\mathcal{R}_0 \leq 1} and the existence and uniqueness of an endemic equilibrium when ${\mathcal{R}_0 >1 }${\mathcal{R}_0 >1 } . We also prove the global asymptotic stability of the endemic equilibrium for a differential susceptibility and staged progression infectivity model, when ${\mathcal{R}_0 >1 }${\mathcal{R}_0 >1 } . Our results encompass and generalize those of Hyman and Li (J Math Biol 50:626–644, 2005; Math Biosci Eng 3:89–100, 2006).     

Studies on a continuous recycle reactor for a lipase-catalysed processing of ricebran oil

The authors have developed a continuous recycle reactor which efficiently performs emulsion type enzymatic reactions. The reactor column is filled with immobilised lipase and the reactions are effected by pumping the pre-prepared oil-water emulsion through the bottom of the reactor. A part of the product was recycled back and this type of recycling greatly improves the productivity of fatty acid compared to continuous once-through reactor without recycling. The recycle reactor could be continuously run for 35 days without decrease in conversions. The performance of the reactor was interpreted by a model and the theoretical conversion was compared with the experimental data.List of Symbols F _AO mol/min feed rate - K _M g/l Michaelis constant - R recycle ratio - r ₅ mol/(ml · min) reaction rate - S ₀ g/l initial substrate concentration - V _max mol/(ml · min) maximum reaction velocity - V _R l void volume of the reactor - x _s fractional conversion - Standard deviation      

Life history of the nonnative convict cichlid <Emphasis Type="Italic">Amatitlania nigrofasciata</Emphasis> in the Haebaru Reservoir on Okinawa-jima Island,Japan

Life history parameters associated with reproductive biology, age, and growth of the convict cichlid (also known as the zebra cichlid) Amatitlania nigrofasciata, which was introduced into the Haebaru Reservoir on Okinawa-jima Island, were estimated using 437 specimens that ranged from 13.7 to 82.9 mm standard length (SL). Lengths of females at first maturity (SL) and 50% maturity (L ₅₀) were estimated to be 32.2 and 37.3 mm SL, respectively. The spawning period continued throughout the year, with a peak spawning cycle from March to May 2006–2007. Observations of postovulatory follicles and tertiary yolk stage oocytes indicate that convict cichlids spawn multiple times within a year. Female cichlids that hatched during the peak spawning seasons matured after October of the same year. Batch fecundity of females (32.2–61.2 mm SL) ranged from 65 to 345 (mean ± SD = 155 ± 63). Opaque zones along the outer margins of otoliths formed annually. The maximum age of male and female cichlids was 3 years. The von Bertalanffy growth formulae (VBGF) were expressed as L_t = 57.4( 1 - e ^{- 0.78( t + 0.91 )} ) {{\hbox{L}}_{\rm{t}}}{ = 57}{.4}\left( {1 - {e^{ - 0.78\left( {t + 0.91} \right)}}} \right) for females and L_t = 69.5( 1 - e ^{- 1.07( t + 0.24 )} ) {{\hbox{L}}_{\rm{t}}}{ = 69}{.5}\left( {1 - {e^{ - 1.07\left( {t + 0.24} \right)}}} \right) for males. Males grew larger than females beginning from the first year. Certain life history characteristics, such as year-round spawning and early maturation, probably contributed to the successful establishment of the convict cichlid, and this species in particular is thought to adapt and become established quickly upon introduction to freshwater systems on Okinawa-jima Island.     

Nitric oxide promotes in vitro organogenesis in <Emphasis Type="Italic">Linum usitatissimum</Emphasis> L.

The effects of nitric oxide (NO) on caulogenesis, shoot organogenesis and rhizogenesis from hypocotyl explants of Linum usitatissimum were investigated. Exogenously supplied NO donors, 5 μM sodium nitroprusside (SNP), 2 μM S-nitroso-N-acetylpenicillamine (SNAP) and 2 μM 3-morpholinosydnonimine (SIN-1), significantly promoted shoot differentiation from the hypocotyl explants of L. usitatissimum excised from its in vitro raised seedlings. Potassium ferrocyanide, a structural analogue of SNP, lacking NO group, did not promote shoot organogenesis. Likewise, products of NO, \textNO₂ ^- {\text{NO}}_{2}^{ - } and \textNO₃ ^- {\text{NO}}_{3}^{ - } supplied as 5 μM NaNO₂ and 5 μM NaNO₃ did not enhance shoot differentiation. Another source of NO, a mixture of sodium nitrite (SN) provided along with ascorbic acid (AsA), also caused significant promotion in the average number of shoots per responding explant. SNP also augmented the rhizogenic response of the microshoots in terms of percentage of responding explants, number of roots per responding explant and average root length. The NO scavengers, 2-(4-carboxy-phenyl)-4, 4, 5, 5-tetramethylimideazoline-1-oxyl-3-oxide (cPTIO) or methylene blue (MB), provided along with SNP, SNAP, SIN-1 or SN + AsA, at concentrations equimolar to the optimum concentration of the donors, reversed the promotory influence, thereby, confirming the role of NO in promotion of in vitro morphogenesis. However, NO scavengers individually did not affect the observed morphogenic processes. Morphological and histological studies of hypocotyl segments cultured on BM or BM + SNP for 4, 8 and 12 days demonstrated that SNP enhanced shoot differentiation by inducing a higher number of shoot primordia, each of which develops into a single shoot.     

Reactive oxygen species production and antioxidative defense system in pea root tissues treated with lead ions: the whole roots level

The lead absorbed by the roots induce oxidative stress conditions through the Reactive oxygen species (ROS) production for the pea plants cultivated hydroponically for 96 h on a Hoagland medium with the addition of 0.1 and 0.5 mM of Pb(NO₃)₂. The alterations in \textO₂ ^{- ·} {\text{O}}_{2}^{ - \cdot } and H₂O₂ concentrations were monitored spectrophotometrically which show a rapid increase in \textO₂ ^{- ·} {\text{O}}_{2}^{ - \cdot } production during the initial 2 h, and in case of H₂O₂, during the eighth hour of cultivation. The level of ROS remained higher at all the time points for the roots of the plants cultivated with Pb²⁺ and it was proportional to metal concentration. The production of \textO₂ ^{- ·} {\text{O}}_{2}^{ - \cdot } and H₂O₂ was visualized by means of fluorescence microscope technique. They are produced in nonenzymatic membrane lipid peroxidation and its final product is Malondialdehyde, the level of which increased together with the level of H₂O₂. As stress intensity raised (duration of treatment and Pb²⁺ concentration), so did the activities of superoxide dismutases, catalase and ascorbate peroxidase antioxidative enzymes and of low-molecular antioxidants, particularly glutathione (GSH), homoglutathione (h-GSH) and cysteine substrate toward their synthesis. The root cells redox state (GSH/GSSG) dropped proportionally to lead stress intensity.     

Gene stacking strategies with doubled haploids derived from biparental crosses: theory and simulations assuming a finite number of loci

Recent progress in genotyping and doubled haploid (DH) techniques has created new opportunities for development of improved selection methods in numerous crops. Assuming a finite number of unlinked loci (ℓ) and a given total number (n) of individuals to be genotyped, we compared, by theory and simulations, three methods of marker-assisted selection (MAS) for gene stacking in DH lines derived from biparental crosses: (1) MAS for high values of the marker score (T, corresponding to the total number of target alleles) in the F₂ generation and subsequently among DH lines derived from the selected F₂ individual (Method 1), (2) MAS for augmented F₂ enrichment and subsequently for T among DH lines from the best carrier F₂ individual (Method 2), and (3) MAS for T among DH lines derived from the F₁ generation (Method 3). Our objectives were to (a) determine the optimum allocation of resources to the F₂ ( n₁^* \, n_{1}^{*} ) and DH generations (n - n₁^* ) (n - n_{1}^{*} ) for Methods 1 and 2 by simulations, (b) compare the efficiency of all three methods for gene stacking by simulations, and (c) develop theory to explain the general effect of selection on the segregation variance and interpret our simulation results. By theory, we proved that for smaller values of ℓ, the segregation variance of T among DH lines derived from F₂ individuals, selected for high values of T, can be much smaller than expected in the absence of selection. This explained our simulation results, showing that for Method 1, it is best to genotype more F₂ individuals than DH lines ($ n_{1}^{*} :n > 0.5 $ n_{1}^{*} :n > 0.5 ), whereas under Method 2, the optimal ratio n₁^* :n n_{1}^{*} :n was close to 0.5. However, for ratios deviating moderately from the optimum, the mean [`(X)] \overline{X} of T in the finally selected DH line ( T<sub>\textDH</sub><sup>*</sup> T_{\text{DH}}^{*} ) was hardly reduced. Method 3 had always the lowest mean [`(X)] \overline{X} of T_\textDH^* T_{\text{DH}}^{*} except for small numbers of loci (ℓ = 4) and is favorable only if a small number of loci are to be stacked in one genotype and/or saving one generation is of crucial importance in cultivar development. Method 2 is under most circumstances the superior method, because it generally showed the highest mean [`(X)] \overline{X} and lowest SD of T_\textDH^* T_{\text{DH}}^{*} for the finally selected DH.     

Minichromosome stability induced by partial genome duplication in Arabidopsis thaliana

Two partially reconstructed karyotypes (RK1 and RK2) of Arabidopsis thaliana have been established from a transformant, in which four structurally changed chromosomes (α, β, γ, and δ) were involved. Both karyotypes are composed of 12 chromosomes, 2n = 1￠￠+ 3￠￠+ 4￠￠+ 5￠￠+ a￠￠+ g￠￠ = 12 {2}n = {1}\prime \prime + {3}\prime \prime + {4}\prime \prime + {5}\prime \prime + \alpha \prime \prime + \gamma \prime \prime = {12} for RK1 and 2n = 3￠￠+ 4￠￠+ 5￠￠+ a￠￠+ b￠￠+ g￠￠ = 12 {2}n = {3}\prime \prime + {4}\prime \prime + {5}\prime \prime + \alpha \prime \prime + \beta \prime \prime + \gamma \prime \prime = {12} for RK2, and these chromosome constitutions were relatively stable at least for three generations. Pairing at meiosis was limited to the homologues (1, 3, 4, 5, α, β, or γ), and no pairing occurred among non-homologous chromosomes in both karyotypes. For minichromosome α (mini α), precocious separation at metaphase I was frequently observed in RK2, as found for other minichromosomes, but was rare in RK1. This stable paring of mini α was possibly caused by duplication of the terminal tip of chromosome 1 that is characteristic of RK1.     

AMP-activated protein kinase plays a role in initiating metabolic rate suppression in goldfish hepatocytes

In the present study, we test the hypothesis that AMP-activated protein kinase (AMPK) initiates metabolic rate suppression in isolated goldfish hepatocytes. To accomplish this, we attempted to pharmacologically activate AMPK in goldfish hepatocytes with 5-aminoimidazole-4-carboxamide ribonucleoside (AICAR) and the thienopyridone, A769662, to examine the effects of AMPK activation on eukaryotic elongation factor-2 (eEF2), protein synthesis, and cellular oxygen consumption rate ( [(M)\dot]_{\textO 2} \dot{M}_{{{\text{O}}_{ 2} }} ). Goldfish hepatocytes treated with 1 mM AICAR under normoxic conditions (>200 μM O₂) showed a modest but significant 1.1-fold increase in AMPK phosphorylation, a 7.5-fold increase in AMPK activity, a 1.4-fold increase in eEF2 phosphorylation, and a 24% decrease in [(M)\dot]_{\textO 2} \dot{M}_{{{\text{O}}_{ 2} }} . At physiologically relevant [O₂] (<40 μM O₂), the addition of 1 mM AICAR resulted in only a 13% decrease in cellular [(M)\dot]_{\textO 2} \dot{M}_{{{\text{O}}_{ 2} }} with no change in sensitivity to [O₂] as assessed by estimates of cellular P₅₀ and P₉₀ values. The addition of compound C, a general protein kinase inhibitor, after AICAR incubation did not reverse the effects of AICAR on [(M)\dot]_{\textO 2} \dot{M}_{{{\text{O}}_{ 2} }} in normoxia. Treatment of hepatocytes with ≤200 μM A769662 did not affect AMPK activity, AMPK phosphorylation, eEF2 phosphorylation, or cellular [(M)\dot]_{\textO 2} \dot{M}_{{{\text{O}}_{ 2} }} . These data suggest that A769662 is not an activator of AMPK in goldfish hepatocytes. Although our study provides support for the hypothesis that AMPK plays a role in initiating metabolic rate suppression in goldfish hepatocytes, this support must be viewed cautiously because of the known off-target effects of the pharmacological agents used.     

Studying Genetic Code by a Matrix Approach

Following Petoukhov and his collaborators, we use two length n zero-one sequences, α and β, to represent a length n genetic sequence ((a) || (b)){\alpha\choose\beta} so that the columns of ((a) || (b)){\alpha\choose\beta} have the following correspondence with the nucleotides: C ~ (0 || 0)C\sim{0\choose0} , U ~ (1 || 0)U\sim{1\choose0} , G ~ (1 || 1)G\sim{1\choose1} , A ~ (0 || 1)A\sim{0\choose1} . Using the Gray code ordering to arrange α and β, we build a 2 ⁿ ×2 ⁿ matrix C _n including all the 4 ⁿ length n genetic sequences. Furthermore, we use the Hamming distance of α and β to construct a 2 ⁿ ×2 ⁿ matrix D _n . We explore structures of these matrices, refine the results in earlier papers, and propose new directions for further research.     

Evaluation of Surface Free Energy of Various Fruit Epicarps Using Acid�CBase and Zisman Approaches

Surface free energy (SFE; γ _SV) of 16 fruit epicarps present on the Chilean market was calculated by two approaches: the acid–base and Zisman. The results show that the fruit epicarps were low surface energy since the magnitude of γ _SV falls within a narrow range, between 37 and 44 mJ m^− 2. Zisman approach gave a critical surface tension values, γ _cr lower than the SFE calculated by the acid–base approach. Significant differences in SFE between the fruits may be explained by the variation in the chemical composition of epicuticular waxes. The polar (g^AB_SV\gamma^{\rm AB}_{\rm SV}) and apolar (g^LW_SV\gamma^{\rm LW}_{\rm SV}) components of the SFE were also calculated and a mathematical relation was between both values was found. Values of g^AB_SV\gamma^{\rm AB}_{\rm SV} and g^LW_SV\gamma^{\rm LW}_{\rm SV} could also be associated with the fruit family and the tissue origins in the ovary region. Finally, it has been shown that fruit epicarps exhibited predominantly electron-donator behaviour since $\gamma_{\rm SV}^- > \gamma_{\rm SV}^+$\gamma_{\rm SV}^- > \gamma_{\rm SV}^+. We believe that the results reported here can potentially impact in food engineering because the compatibility of coatings and fruit epicarps depends on the interaction of their respective chemical and physical properties.     

Viscosity of heparin as a function of dielectric constant and of desulfation

The effect of dielectric constant (D) of the solvent on the viscosity of heparin was examined using the relation \documentclass{article}\pagestyle{empty}\begin{document}$ \eta _{{\rm sp}} /c = [\eta ]_\infty (1 + k/\sqrt c) $\end{document}, where [η]_∞ is the shielded intrinsic viscosity obtained by extrapolating \documentclass{article}\pagestyle{empty}\begin{document}$ \eta _{{\rm sp}} /c\,{\rm vs}{\rm . }\,1/\sqrt c ) $\end{document} to infinite concentration, and k is an interaction parameter independent of the dielectric constant of the solvent. This equation was previously reported by the authors⁹ for describing the reduced viscosities of strong polyelectrolytes in salt-free polar solvents. It was found that the [η]_∞ of heparin increases linearly with increasing dielectric constant of the solvent whereas the k values were, within experimental error, independent of D in the range 54.7 < D < 93.2 examined. Graded hydrolysis of heparin from its acid form (heparinic acid) at 57°C resulted in samples of varying degree of desulfation with corresponding decrease in biological activity. It was found that both [η]_∞ and k decrease with increasing desulfation.     

Using scales to clarify the transition from blue-phase to brown-phase fingerling in Notothenia rossii from the South Shetland Islands

Scales and whole otoliths were read for age determination in early stages of Notothenia rossii caught in Potter Cove, South Shetland Islands, in summer seasons 2003–2006 and 2008. The sample comprised blue-phase pelagic fingerlings of 7.0–7.6 cm (TL) of age group 0 year and demersal brown-phase fingerlings/juveniles of 8.5–20.9 cm and predominant age groups 1–2 years. Counting of sclerites facilitated the interpretation of the rings, particularly in the central scale. To clarify two previous issues of controversy, we deduce that the duration of the offshore pelagic blue-phase fingerling stage is less than one year before migration to the nearshore demersal habitat. Furthermore, the first well-defined ring in scales corresponded to the first annulus, while a contiguous ring was a secondary ring sometimes deposited after the first winter during the second year of life, attributable to a shift of habitat from pelagic to demersal. A von Bertalanffy growth curve was computed by combining age/length data of the juvenile phase of N. rossii from this and a previous study at Potter Cove with literature data from the offshore adult population, resulting in the following equation: L_t = 8 6. 9  ( 1- \texte ^{- 0. 0 9 1 (t - 0. 6 6 8 )} ) L_{t} = 8 6. 9 \,\left( { 1- {\text{e}}^{{ - 0. 0 9 1 (t - 0. 6 6 8 )}} } \right) _.     

Influence of {\text{NH - }}{{\text{S}}^\gamma } bonding interactions on the structure and dynamics of metallothioneins

Mammalian metallothioneins ( \textM₇^\textIIMTs {\text{M}}_7^{\text{IIMTs}} ) show a clustered arrangement of the metal ions and a nonregular protein structure. The solution structures of Cd₃-thiolate cluster containing β-domain of mouse β-MT-1 and rat β-MT-2 show high structural similarities, but widely differing structure dynamics. Molecular dynamics simulations revealed a substantially increased number of \textNH - \textS^g {\text{NH - }}{{\text{S}}^\gamma } hydrogen bonds in β-MT-2, features likely responsible for the increased stability of the Cd₃-thiolate cluster and the enfolding protein domain. Alterations in the \textNH - \textS^g {\text{NH - }}{{\text{S}}^\gamma } hydrogen-bonding network may provide a rationale for the differences in dynamic properties encountered in the β-domains of MT-1, -2, and -3 isoforms, believed to be essential for their different biological function.     

HNCA-TOCSY-CANH experiments with alternate <Superscript>13</Superscript>C-<Superscript>12</Superscript>C labeling: a set of 3D experiment with unique supra-sequential information for mainchain resonance assignment

Described here is a set of three-dimensional (3D) NMR experiments that rely on CACA-TOCSY magnetization transfer via the weak ³ \textJ_{\textCa\textCa} ^{ 3} {\text{J}}_{{{\text{C}}\alpha {\text{C}}\alpha }} coupling. These pulse sequences, which resemble recently described ¹³C detected CACA-TOCSY (Takeuchi et al. 2010) experiments, are recorded in ¹H₂O, and use ¹H excitation and detection. These experiments require alternate ¹³C-¹²C labeling together with perdeuteration, which allows utilizing the small ³ \textJ_{\textCa\textCa} ^{ 3} {\text{J}}_{{{\text{C}}\alpha {\text{C}}\alpha }} scalar coupling that is otherwise masked by the stronger ¹J_CC couplings in uniformly ¹³C labeled samples. These new experiments provide a unique assignment ladder-mark that yields bidirectional supra-sequential information and can readily straddle proline residues. Unlike the conventional HNCA experiment, which contains only sequential information to the ^{1 3} \textC^a ^{ 1 3} {\text{C}}^{\alpha } of the preceding residue, the 3D hnCA-TOCSY-caNH experiment can yield sequential correlations to alpha carbons in positions i−1, i + 1 and i−2. Furthermore, the 3D hNca-TOCSY-caNH and Hnca-TOCSY-caNH experiments, which share the same magnetization pathway but use a different chemical shift encoding, directly couple the ¹⁵N-¹H spin pair of residue i to adjacent amide protons and nitrogens at positions i−2, i−1, i + 1 and i + 2, respectively. These new experimental features make protein backbone assignments more robust by reducing the degeneracy problem associated with the conventional 3D NMR experiments.     

A mathematical model of biological evolution

In order to understand generally how the biological evolution rate depends on relevant parameters such as mutation rate, intensity of selection pressure and its persistence time, the following mathematical model is proposed: dN _n (t)/dt=(m _n (t-)N _n (t)+N _n-1(t) (n=0,1,2,3...), where N _n (t) and m _n (t) are respectively the number and Malthusian parameter of replicons with step number n in a population at time t and is the mutation rate, assumed to be a positive constant. The step number of each replicon is defined as either equal to or larger by one than that of its parent, the latter case occurring when and only when mutation has taken place. The average evolution rate defined by is rigorously obtained for the case (i) m _n (t)=m _n is independent of t (constant fitness model), where m _n is essentially periodic with respect to n, and for the case (ii) (periodic fitness model), together with the long time average m of the average Malthusian parameter . The biological meaning of the results is discussed, comparing them with the features of actual molecular evolution and with some results of computer simulation of the model for finite populations.An early version of this study was read at the International Symposium on Mathematical Topics in Biological held in kyoto, Japan, on September 11–12, 1978, and was published in its Procedings.