Disappearance of gas vacuoles in the blue-green alga Microcystis aeruginosa

Abstract In a culture of Microcystis aeruginosa, which had been transferred from a mineral medium into distilled water, the number of gas vacuoles per cell decreased and reached a value of 20% of the control 24 h after transfer. In senescent cells grown on a mineral agar for several weeks, the gas vacuoles also disappeared. The disappearance of the gas vacuoles may be a response to a nutrient deficiency in both cultures.     

Counting of gas vacuoles by electron microscopy in lysates and purified fractions ofmicrocystis aeruginosa

Summary A microdroplet spray method is described to determine quantitatively the number of gas vacuoles in purified fractions and in total lysates of the blue-green alga,Microcystis aeruginosa.It is found that 6.45×10¹² gas vacuoles of 370 nm length make up one mg of protein. This supports the idea that the entire organelle is made up by a single layer of protein only. The gas vacuole number per cell varies with the stage of growth between 3,700 and 11,000.     

Growth components in Allium roots

Summary This paper reports results of a study of root growth, duration of the division cycle, and cell size for different intervals of root length, associated with different phases of growth, namely, from the onset of growth to the achievement of a steady state characterized by a constant rate of growth. The material is roots of Allium cepa.Steady-state kinetics of growth are achieved only when all components of growth reach a constant value. Although the cell flow remains constant from a root length of 8 mm onwards, the steady condition is only observed when the final size of the epidermal cells reaches its maximum and constant value (at about 12 mm of root length).The equation G=24×10^-5 \N\·L is proposed to analyze the growth of Allium roots in its components. G is the growth rate (mm/24 hr), N the number of cells per column in the meristematic ribs, the rate of the cell cycle (the reciprocal of its duration x100), and L the average final size of the epidermal cells (in ).     

Physiological effects of the presence and absence of gas vacuoles in the blue-green alga,Microcystis aeruginosa Kuetz. emend. Elenkin

Physiological evidence was obtained for a light shielding role for gas vacuoles inMicrocystis aeruginosa Kuetz. emend. Elenkin, by comparing photosynthetic oxygen evolution, growth behaviour and pigment composition of cells with intact or collapsed gas vacuoles. The oxygen evolution rates were strongly dependent on cell concentration, a maximum rate for cells with intact gas vacuoles occurring at about 1.4×10⁹ cells/ml and for cells with collapsed gas vacuoles at about 2.5×10⁹ cells/ml. By using light saturation curves for oxygen evolution, it was estimated that at low light intensities up to 30% of the photosynthetically useable light was shielded at a cell concentration of 6×10⁸ cells/ml. Collapsing the gas vacuoles twice daily did not alter the initial growth rate of the cultures, but enabled them to reach a higher final cell density. Collapsing of gas vacuoles during growth for about four generations resulted in a lower level of all acetone soluble pigments with a greater relative reduction in carotenoids than in chlorophyll a. Collapse of the gas vacuoles does not alter the cell volume. Various optical interactions which could account for light shielding are discussed.     

Growth characteristics of Uranoscopus scaber Linnaeus 1758, inhabiting the Iskenderun Bay (Northeastern Mediterranean)

This study aimed to determine the age and some growth characteristics of Atlantic stargazer (Uranoscopus scaber) from Iskenderun Bay (Northeastern Mediterranean). For this purpose, a total of 150 Atlantic stargazer ranging in size from 9.1 to 28.0 cm in total length (weight: 11.7–345.7 g) were collected as by-catch from a commercial trawl fishing boat at a depth of 80–100 m between May 2015 and January 2016. The bottom trawl gear used was equipped with a 44 mm stretched mesh size net at the cod-end. The percentage of females and males were 46.7% and 53.3% respectively. The total length–weight relationships equation with coefficient of determination (R²) were found as W = 0.011*TL^3.131, R² = .9728, for all individuals, W = 0.015*TL^3.021, R² = .9512 for females and W = 0.0102*TL^3.136, R² = .9553 for males. By using the von Bertalanffy equation, the growth parameters of Atlantic stargazer were estimated to be L_∞ = 42.35 cm, k = 0.098, t₀ = −1.8474 for all individuals; L_∞ = 36.92 cm, k = 0.138, t₀ = −1.2693 for females and L_∞ = 38.77 cm, k = 0.1, t₀ = −2.334 for males. In this study, age reading was done by two independent readers and index of average percentage error (IAPE) was calculated as 6.1%. The highest condition factor value calculated as 1.81 in the age group 6 and the lowest condition factor value was calculated as 1.48 in the age group 1.     

Influence of operational parameters in the flocculation and production ofLactobacillus plantarum

Summary The influence of different operational parameters, such as the dilution rate (D) and the bleeding rate (B), in the production of a flocculent strain ofLactobacillus plantarum was studied. The effect of the dilution rate was demonstrated to be related to the lactic acid concentration inside the reactor. The effect of the bleeding rate was shown to be critical in the stabilization of the operation (due to a better pH control). It also allowed a continuous recovery of cells outside the reactor. Viability testing of the lactic starter cultures showed that operation with cell purge increased the viability of the starter cultures obtained.Nomenclature B Bleeding rate, h^–1 - D Dilution rate, h^–1 - F Feed flow rate, L h^–1 - I Feed velocity, m h^–1 - Specific growth rate, h^–1 - v Lactic acid specific productivity, g g^–1 h^–1 - P Product concentration (lactic acid), g L^–1 - P _out Product concentration leaving the system, g L^–1 - Q _b Bleeding flow rate, L h^–1 - R Recirculation velocity, m h^–1 - S Substract concentration, g L^–1 - t Time, h - T _p Time of ascensional flow (length of the column/total ascensional velocity), h - T _r Residence time (1/D), h - V Volume of the reactor, L - X Cell concentration, g L^–1 - X _out Cell concentration leaving the system, g L^–1     

Theoretical analysis of the baker's yeast production: An experimental verification at a laboratory scale

The scale-down procedure seems an adequate tool in the design, optimization and scale-up fermentation processes. The first step in this procedure is a theoretical analysis, called process analysis, which is based on characteristic times of the mechanisms which may influence the performance of the bioreactor. This analysis must give information about the behaviour of large and small scale fermentation processes. At a small scale a verification of the results of such an analysis of the fed-batch baker's yeast production is carried out.In this paper a comparison of calculated and measured characteristic times of liquid mixing and mass transfer is presented. It was concluded that the literature correlations give a rough estimation of the characteristic times and can be used in the process analysis. Depending on the kind of sparger, the medium and the scale of the reactor, more knowledge is needed about bubble coalescence in fermentation media.The volumetric oxygen transfer coefficient increased when the biomass concentration increased. Probably this is caused by the interaction between biomass and the anti-foaming agent used.List of Symbols C kg/m³ concentration - D m diameter - m²/s effective dispersion coefficient - d m holes of the sparger - g m/s² gravitational acceleration - H m height - k _L a s^–1 volumetric mass transfer coefficient based on the liquid volume - L m length - m kg/kg gas liquid distribution coefficient - OTR kg/(m³ · s) oxygen transfer rate - OUR kg/(m³ · s) oxygen uptake rate - t s time - ^s m/s superficial gas flow rate - m length - s time constant - _g m³/s gas flow rate Indices 0 value at t=0 - cal calculated - e value at t=t (end) - g gas phase - in flow going to the fermentor - l liquid phase - m mixing - mt mass transfer - O ₂ oxygen - out flow coming out the fermentor     

Age,growth, reproduction and status of resource development of Ptychidio jordani,a critically endangered freshwater fish in the Hongshui River,China

The age, growth, reproduction and resource development status of Ptychidio jordani, as a critically endangered freshwater fish in the Hongshui River, China, was studied in this work. A total of 525 specimens were collected monthly using the cages and gillnets from October 2021 to September 2022 in the Hongshui River. The scale was used for age determination, and the maximum age for both female and male was estimated to be 5 years and 3 years, respectively. Female and male P. jordani showed different growth patterns, which were expressed as L_t = 261.3 (1-e^{−0.4885(t−0.1476)}) and L_t = 251.2 (1-e^{−0.4758(t+0.9643)}), respectively. The overall sex ratio was 1:0.47 (female:male). Female attained sex maturity at 2.34 years (192 mm body length). Month variation of the gonad somatic index indicated that the spawning period occurred from April to October. The absolute fecundity was estimated at 9046 ± 3434 eggs per individual, and the relative fecundity was 38.08 ± 15.77 eggs per gram. The exploitation rate of female and male was 0.233 and 0.495, which indicated that P. jordani was not overfishing. This study provided data on the key life-history traits of P. jordani, which has not been known previously and is essential for conservation strategy and policy development.     

The viability to a wall shear stress and propagation of <Emphasis Type="Italic">Bifidobacterium longum</Emphasis> in the intensive membrane bioreactor

Bifidobacterium longum grew at 65 L pilot scale of the membrane bioreactor (MBR), externally fitted with ceramic membrane (0.7 m²). Cell mass at the MBR reached 22.18 g L⁻¹ as dry cell weight in 12 h, which is 8.44 times higher than cell mass attained at the vial culture. The growth rate in the vial culture was μ = 0.385 h⁻ and at the batch culture was μ = 1.13 h⁻ in the exponential period and μ = 0.31 h⁻¹ in the stationary period. In the fed-batch mode was μ = 1.102 h⁻¹ for 6 h with inoculation and declined to μ = 0.456 h⁻¹ with feeding of feed medium. The growth rate at the MBR was μ = 0.134 h⁻¹. The number of viable cells was 6.01 × 10¹² cfu L⁻¹ at the batch culture, but increased to 1.15 × 10¹⁴ cfu L⁻¹ at the MBR culture. The specific growth rate of viable cell number (colony-forming units per liter, per hour) improved by 6.01 times from the batch to the MBR culture. The wall shear stress mainly generated by the pump, and the membrane incorporated into the MBR was controlled during the cultivation at the MBR. The viability of B. longum declined to under 10% in the first 2 weeks of the 4-week stability test (40°C) as B. longum was exposed to over wall shear stress 713 Pa, but the viability improved to 30–40% in wall shear stress of 260 Pa or STR culture. The loss in the cell viability can be saved by managing with wall shear stress during the cultivation at the MBR.     

Age, growth and reproduction of the humpback red snapper Lutjanus gibbus off Ishigaki Island, Okinawa

The humpback red snapper Lutjanus gibbus (Lutjanidae) is an important species for fisheries in the Kagoshima and Okinawan region of Japan. The present study estimated the age, growth and reproduction of this lutjanid species in the waters around Ishigaki Island, southern part of Okinawa. An opaque zone was formed on the otolith every year, and this formation correlated with their spawning season; these zones were identified as annual rings. Maximum ages of 21 and 24 years were observed for males and females, respectively. The von Bertalanffy growth parameters clarifying the age–fork length relationship were as follows: L _∞ = 390.5 mm, K = 0.210 year⁻¹ and t ₀ = −1.88 year for males, and L _∞ = 303.4, K = 0.256 year⁻¹ and t ₀ = −3.05 year for females. The main spawning season was estimated as between May and October, since greater values of gonadosomatic index for females as well as maturation oocytes and/or postovulatory follicles were observed during those six months.     

Modelling the distribution of the invasive Ziziphus mauritiana along road corridors in Zimbabwe

We investigate how human fruit consumption affects the spread of the alien invasive Ziziphus mauritiana along road corridors in northern Zimbabwe. A field survey was conducted to identify and map Z. Mauritiana stems at 25 randomly located 6‐ha plots placed along two main roads connecting the Zambezi Valley to the Highveld region. The L‐function was used to test for evidence of significant spatial clustering of the stems. The inhomogeneous point model fitted by maximum likelihood was also applied to check whether distance from the road explains variation in the number of stems per unit area. Finally, a t test was executed on log‐transformed abundance data to test for significant differences in the mean number of saplings and adults between the Zambezi Valley and the Highveld. Results of the L‐function indicated that Z. mauritiana had a clustered and linear distribution along roads both in the Zambezi Valley and Highveld regions. Results of the t test showed that the mean number of saplings per plot in the Zambezi Valley (μ₁ = 275) was three times higher than in the Highveld (μ₂ = 78), with p < 0.01. The mean number of adult trees was also significantly higher in the Zambezi Valley than in the Highveld.     

Effects of light regime and oxygen profile on transformation of oxolinic acid in pond sediment

This study investigated the effects of light (visible light - 5800 lux, 24 h) or dark regime and aerobic or anaerobic condition on the decay of added oxolinic acid (OA) at 5, 10 and 20 mg L⁻¹ in eel pond sediment. An asymptotic decaying exponential model C_t = C_min + C_o × exp (−k × t) was used to facilitate quantitative approach to OA transformation, where C_t is the concentration of OA after t days, C_min the estimated level-off concentration of OA residue, C_o the concentration of added OA and k the decaying coefficient. OA decayed faster under light (C_min = 4.6 mg L⁻¹) than under dark (C_min = 7.8 mg L⁻¹) and also decayed faster under aerobic (C_min = 4.0 mg L⁻¹) than under anaerobic condition (C_min = 8.5 mg L⁻¹). C_min increased with C_o. Sundrying and tilling eel pond bottom should be able to reduce OA residue significantly.     

Reproductive biology of Metaschistura cristata in northeastern Iran

The study purpose was to investigate the reproductive biology, growth and length‐weight of the Turkmenian crested loach, Metaschistura cristata, in the Radkan River of northeastern Iran. Age and growth are described for 1029 specimens from January 2009 to December 2010. The sex ratio was 1:1. Maximum age, based on opercula readings, was 6⁺ years for both sexes. Specimens ranged in size from 24 to 98 mm total length and weighed from 0.08 to 7.32 g. The length‐weight relationships were described as W = 0.005166 TL^3.225 (R² = 0.97) for males and W = 0.006192 TL^3.125 (R² = 0.97) for females. Growth was expressed in length and the von Bertalanffy growth parameters were estimated as L_∞ = 354.9 mm, k = 0.0038, t₀ = ?26.82 for males and L_∞ = 339.0 mm, k = 0.0043, t₀ = ?24.88 for females. Growth performance indexes were also estimated as Φ′ = 6.17 for males and Φ′ = 6.20 for females. The gonadosomatic index showed that peak reproduction occurred during April and June, with highest average values of 2.1% for males and 25.3% for females in May. Oocyte diameter ranged from 0.09 to 1.58 mm, with a mean value of 0.54 ± 0.42 mm.     

Indirect shoot organogenesis from leaves of Dieffenbachia cv. Camouflage

A novel protocol for indirect shoot organogenesis of Dieffenbachia cv. Camouflage was established using leaf explants excised from in vitro shoot cultures. The frequency of callus formation reached 96% for explants cultured on Murashige and Skoog (1962) basal medium supplemented with 5 μM thidiazuron and 1 μM 2,4-dichlorophenozyacetic acid. The number of shoots regenerated was high, with up to 7.9 shoots produced per callus cultured on basal medium supplemented with 40 μM N ⁶-(Δ²-isopentenyl)adenine and 2 μM indole-3-acetic acid. Regenerated shoots rooted well in a soilless substrate, acclimatized ex vitro at 100%, and grew vigorously under shaded greenhouse conditions. Somaclonal variations in leaf variegation, color, and morphology have been observed in regenerated plants.     

The DNA,RNA and protein composition of the cyanobacterium Anacystis nidulans grown in light- and carbon dioxide-limited chemostats

The DNA, RNA and protein content of the cyanobacterium Anacystis nidulans was determined in light-limited and carbon dioxide-limited chemostat cultures over the dilution rate range, D=0.02 h^-1 to 0.19 h^-1. The macromolecular contents as a percentage of the dry weight and on a per cell basis varied significantly as a function of organism growth rate and the nature of the growth conditions. For both limitations the RNA content per cell increased [20–55 fg RNA (cell)^-1] with increasing dilution rate and also showed an increase as a percentage of the dry weight. The DNA content as a percentage of the dry weight showed a 2-fold decrease with increasing dilution rate over the range examined. On a per cell basis DNA reached a peak at D=0.1 h^-1 [4.5 fg DNA (cell)^-1] for light-limited organisms and at D=0.08 h^-1 [8.0 fg DNA (cell)^-1] for carbon dioxide-limited organisms. The q _RNA increased with increasing dilution rates over the complete growth rate range examined whilst q _DNA reached a maximum at D=0.09 to 0.10 h^-1. The protein content as a percentage of the dry weight was greater in CO₂-limited organisms than light-limited organisms but in both cultures declined as the dilution rate was increased above D=0.10 h^-1.     

Presence of activated lymphocytes in the urine of patients with superficial bladder cancer after intravesical immunotherapy with bacillus Calmette-Guérin

Summary To study the mode of action of intravesical bacillus Calmette-Guérin (BCG) immunotherapy in the prevention and cure of superficial bladder cancer, flow-cytofluorometric analysis of the cellular immunological reaction in the urine of patients was performed. Fresh urinederived leucocytes were obtained from eight patients before (t ₀) and 24 h (t ₂₄) and 48 h (t ₄₈) after repeated intravesical BCG instillations (at least 5 instillations). For two patients urine-derived leucocytes were investigated at the first BCG instillation. The number of leucocytes in the urine was markedly increased 24 h after repeated BCG instillations, indicating a local cellular immunological reaction induced by BCG. The mean number of cells per milliliter of urine at that time was 2.9×10⁶±3.6×10⁶ (n = 8). These leucocytes consisted mainly of granulocytes (75±11%,n = 8). In addition monocytes/macrophages (4±2%,n = 8) and T lymphocytes were present (1±1%,n = 5). The relative increase of monocytes/macrophages in the urine after BCG application tended to be higher compared to the other leucocyte subtypes. As T lymphocytes may play an important role in the BCG-mediated antitumour activity, subsets of lymphocytes were further characterized att ₀,t ₂₄, andt ₄₈ after repeated BCG instillations. The lymphocyte population consisted mainly of T cells (86% CD3⁺,t ₀). Most of the T cells were CD4⁺ (helper/inducer) and were significantly decreased at 48 h (62±9% att ₀ vs 49±6% att ₄₈). Lymphocytes partly expressed HLA-DR antigens (44%,t ₀). The percentage of lymphocytes with interleukin-2 (IL-2) receptors (CD25⁺) was significantly increased at 24 h and 48 h, compared to pre-instillation values (19±11% and 10±4% vs 3±3% respectively). Natural killer cells (CD 16⁺ and/or CD56⁺) and B cells (CD 19⁺) were less numerous (10% and 19% att ₀ respectively). After the first BCG instillation the increase in the number of leucocytes in urine seemed to be less compared to the numbers after repeated BCG instillations. Lymphocytes could not be detected in the urine collected before or after the first BCG instillation. In conclusion, we demonstrated the presence of considerable numbers of leucocytes in the urine 24 h after repeated BCG instillations, i.e. shortly after immunological activation. The antigen expression of the lymphocytes suggested that they may represent the lymphocytes in the bladder wall. Expression of HLA-DR and IL-2 receptors on lymphocytes indicated activation of T cells by the intravesical BCG treatment. These leucocytes may be useful for functional studies, which are essential to elucidate the actual effector mechanism(s) in the mode of action of BCG against superficial bladder cancer in man.     

Startup of anaerobic biofilm fluidized bed reactors on molasses and phenol under various conditions

Fluidized sand bed anaerobic biofilm reactors were operated in parallel to study the effects of inoculum, loading, residence time and carrier type on the startup dynamics for the degradation of molasses and phenol. Degradation rates generally depended most directly on concentrations rather than on other operating variables. Residence times did not appear to directly influence startup. Short residence times and high loadings gave the highest specific activities for both substrates. The type of inoculum was found to be most important for the molasses system, and inoculation on fresh carrier was found to be better than reinoculation. The two times higher specific biomass retention on Siran porous glass gave essentially the same degradation rates on a volume basis.List of Symbols L kg/h loading of reactor - M kg/kg biomass per carrier mass - Red. % reduction of feed concentration due to degradation - R kg/(m³ · h) reaction rate - S kg/m³ substrate concentration in reactor and effluent - S ₀ kg/m³ substrate concentration in feed - t h time     

Age and growth of the blackchin guitarfish Glaucostegus cemiculus (Geoffroy Saint-Hilaire, 1817) from Iskenderun Bay (Northeastern Mediterranean)

Between September 2010 and June 2012, a total 291 (166 females and 125 males) blackchin guitarfish Glaucostegus cemiculus were captured by a commercial bottom trawler (F/V Coşkun Reis) in Iskenderun Bay, Turkey (northeastern Mediterranean Sea). The total length (L) and total weight (W) of the female and male guitarfish ranged between 32.0–165.0 cm and 88 g–16.68 kg, and 34.3–128.3 cm and 112 g–6.00 kg, respectively. Vertebral age estimates ranged from 0 to 8 years for females and 0 to 5 years for males. The growth models of von Bertanlanffy and Gompertz were fitted to the length at age data using the nonlinear regression method. Model selection was based on the values of the residual standard error and the Akaike's information criterion corrected for small sample size (AIC_C) associated with each fit. The von Bertalanffy growth model provided the best fitting growth curves for each sex with parameters reaching L_∞ = 187.17 cm, K = 0.195 year^-1, t₀ = −1.38 year for females, and L_∞ = 144.85 cm, K = 0.321 year^-1, t₀ = −1.13 year^-1 for males. The W–L relationship parameters did not differ significantly between sexes, the estimated values of a and b were 0.0018 and 3.11, respectively. By using these values of a and b, and also respective estimates of L_∞, the values of W_∞ were obtained as 20.53 kg for females and 9.25 kg for males. The overall percentage ratios of females and males in the samples were 57% and 43% respectively.     

Age and growth of Schizopygopsis younghusbandi younghusbandi in the Yarlung Zangbo River in Tibet, China

Schizopygopsis younghusbandi younghusbandi is an endemic species whose distribution is restricted to the middle reaches of the Yarlung Zangbo River, being one of the most important commercial fishes in this area. Age and growth of 606 specimens captured between October 2002 and April 2005 were studied. The range in standard length (L) was 65.7–387.3 mm and total weight (W) was 3.3–772.0 g. The relationship between L and W was W = 0.000909L^2.2493 for males and W = 0.000259L^2.4781 for females. Age, determined from anal scales and lapillus otoliths, ranged from 3 to 18 years. The parameters of von Bertalanffy growth functions, estimated by back-calculated length, were L_￥ = 442.7mm  LL_\infty = 442.7mm\;L, k = 0.0738 year⁻¹ and t ₀  = −1.4 year for males, and L_￥ = 471.4mm  LL_\infty = 471.4mm\;L, k = 0.0789 year⁻¹ and t ₀ = 0.2 year for females. Males and females exhibited statistically significant differences in growth. χ ²-test indicated that von Bertalanffy growth functions could well describe the growth of S. y. younghusbandi. The longevities were 39.2 and 38.2 years for males and females, respectively. Growth inflexion points were 10.2 and 12.0 years for males and females, respectively, but 84.8% of the captures were at the smaller ages. So conservation and management schemes for this population should be considered urgently. In addition, we found that populations from the upstream of the Lhasa River, the downstream of the Lhasa River and the middle reaches of the Yarlung Zangbo River showed statistically significant differences in growth patterns.     

Comparison of tomato root distributions by minirhizotron and destructive sampling

Calibration of minirhizotron data against root length density (RLD) was carried out in a field trial where three drip irrigation depths: surface (R0) and subsurface, 0.20 m (RI) and 0.40 m depth (RII) and two processing tomato cultivars: `Brigade' (CI) and `H3044' (CII) were imposed. For each treatment three minirhizotron tubes were located at 10, 37.5 and 75 cm of the way from one plant row to the next. Roots intersecting the minirizotrons walls were expressed as root length intensity (L _a) and number of roots per unit of minirhizotron wall area (N _ra). Root length density (RLD) was calculated from core samples taken for each minirhizotron tube at two locations: near the top of the minirhizotron (BI) and 15 cm apart from it, facing the minirhizotron wall opposite the plant row (BII). Minirhizotron data were regressed against RLD obtained at BI and BII and with their respective means. The results show that for all the situations studied, better correlations were obtained when RLD was regressed with L _a than with N _ra. Also was evident that the relationship between L _a and RLD was strongly influenced by the location of soil coring. RLD was correlated with L _a trough linear and cubic equations, having the last ones higher determination coefficients. For instance at 10 cm from the plant row when values from the top layer (0–40 cm) were analysed separately, L _a was significantly regressed with RLD measured at BII and described by the equations: RLD = 0.5448 + 0.0071 L _a (R ² = 0.51) and RLD = 0.4823 + 0.0074L _a + 8×10^–5 L _a ² – 5×10^–7 L _a ³ (R ² = 0.61). Under the 40 cm depth the highest coefficients of determination for the linear and cubic equations were respectively 0.47 and 0.88, found when L _a was regressed with RLD measured at BI. For minirhizotrons located at 75 cm from the plant row and for location BI it was possible to analyse jointly data from all depths with coefficients of determination of 0.45 and 0.59 for the linear and cubic equations respectively.