Thermodynamical interpretation of an adaptive walk on a Mt. Fuji-type fitness landscape: Einstein relation-like formula holds in a stochastic evolution

We have theoretically studied the statistical properties of adaptive walks (or hill-climbing) on a Mt. Fuji-type fitness landscape in the multi-dimensional sequence space through mathematical analysis and computer simulation. The adaptive walk is characterized by the "mutation distance" d as the step-width of the walker and the "population size" N as the number of randomly generated d-fold point mutants to be screened. In addition to the fitness W, we introduced the following quantities analogous to thermodynamical concepts: "free fitness" G(W) is identical with W+T x S(W), where T is the "evolutionary temperature" T infinity square root of d/lnN and S(W) is the entropy as a function of W, and the "evolutionary force" X is identical with d(G(W)/T)/dW, that is caused by the mutation and selection pressure. It is known that a single adaptive walker rapidly climbs on the fitness landscape up to the stationary state where a "mutation-selection-random drift balance" is kept. In our interpretation, the walker tends to the maximal free fitness state, driven by the evolutionary force X. Our major findings are as follows: First, near the stationary point W*, the "climbing rate" J as the expected fitness change per generation is described by J approximately L x X with L approximately V/2, where V is the variance of fitness distribution on a local landscape. This simple relationship is analogous to the well-known Einstein relation in Brownian motion. Second, the "biological information gain" (DeltaG/T) through adaptive walk can be described by combining the Shannon's information gain (DeltaS) and the "fitness information gain" (DeltaW/T).     

On the progenitor cell migration velocity

An attempt is presented to extract cell kinetic information from histomorphological features. It is applicable to rapidly proliferating tissues like the intestinal epithelium. Each replicating tissue has an origin where cells are formed and a periphery toward which cells migrate. The migration path along which they move is denominated as tissue radius on which all cell positions are mapped. Cell migration on the radius is associated with cell proliferation at tissue origin. Each mitosis there is associated with the displacement of all cells distal to it by one cell position. The more mitoses positioned between a cell and tissue origin, the greater its migration velocity. It is possible therefore to derive the cell migration velocity v(x) from the cumulative mitotic distribution on the radius, N(x). v(x) = N(x)/tm (tm = mitotic time). In this form v(x) represents also cell production at any point on the radius and may serve for the computation of other cell kinetic parameters like generation time. These arguments are illustrated on the rat incisor tooth inner enamel epithelium which has been studied in the normal and rapidly erupting tooth.     

Increased bone calcium following endurance exercise in the mature female rat

In order to determine the effects of exercise on the calcium status of selected axial and appendicular bones of mature rats, female Sprague-Dawley rats (8-9 mo.) were divided into three groups including, two months (E2, n = 8) or four months (E4, n = 9) of exercise, and four month sedentary controls (S, n = 10). Exercise consisted of treadmill running for 1 hr/day, 5 days/wk at a speed of 14.1 m/min and 8 degrees elevation. After sacrifice all femurs, tibia/fibula complexes, ribs (T7), and vertebrae (T7) were excised, cleaned, weighed and measured for length and volume. After freeze-drying and bone hydrolysis in 5N HCl, total bone calcium contents and concentrations were determined spectrophotometrically. The acid soluble, appendicular bone calcium contents of the E4 group were significantly greater than S for the femur and tibia respectively: E4 = 159.78 +/- 3.44 mg (mean +/- SEM), 129.46 +/- 4.87 mg; S = 140.03 +/- 5.04 mg, 110.40 +/- 4.71 mg. Bone calcium concentration (mg/g dry bone) also was significantly greater in the tibia/fibulas, ribs and vertebrae of the E4 group than the S group. With respect to other training-induced effects, the oxygen carrying capacity of the blood, as well as the heart and lung DNA and protein concentrations did not change after four months of exercise training. Within four months, moderate exercise can increase the calcium deposition in the bones of mature, female rats.     

Site-directed mutagenesis improves catalytic efficiency and thermostability of Escherichia coli pH 2.5 acid phosphatase/phytase expressed in Pichia pastoris

Escherichia coli pH 2.5 acid phosphatase gene (appA) and three mutants were expressed in Pichia pastoris to assess the effect of strategic mutations or deletion on the enzyme (EcAP) biochemical properties. Mutants A131N/ V134N/D207N/S211N, C200N/D207N/S211N, and A131N/ V134N/C200N/D207N/S211N had four, two, and four additional potential N-glycosylation sites, respectively. Extracellular phytase and acid phosphatase activities were produced by these mutants and the intact enzyme r-AppA. The N-glycosylation level was higher in mutants A131N/V134N/D207N/S211N (48%) and A131N/V134N/ C200N/D207N/S211N (89%) than that in r-AppA (14%). Despite no enhancement of glycosylation, mutant C200N/ D207N/S211N was different from r-AppA in the following properties. First, it was more active at pH 3.5-5.5. Second, it retained more (P < 0.01) phytase activity than that of r-AppA. Third, its specific activity of phytase was 54% higher. Lastly, its apparent catalytic efficiency kcat/Km for either p-nitrophenyl phosphate (5.8 x 10(5) vs 2.0 x 10(5) min(-1) M(-1)) or sodium phytate (6.9 x 10(6) vs 1.1 x 10(6) min(-1) M(-1)) was improved by factors of 1.9- and 5.3-fold, respectively. Based on the recently published E. coli phytase crystal structure, substitution of C200N in mutant C200N/D207N/S211N seems to eliminate the disulfide bond between the G helix and the GH loop in the alpha-domain of the protein. This change may modulate the domain flexibility and thereby the catalytic efficiency and thermostability of the enzyme.     

Realizing Reversible Conversion‐Alloying of Sb(V) in Polyantimonic Acid for Fast and Durable Lithium‐ and Potassium‐Ion Storage

Finding suitable electrode materials for alkali‐metal‐ion storage is vital to the next‐generation energy‐storage technologies. Polyantimonic acid (PAA, H₂Sb₂O₆ · nH₂O), having pentavalent antimony species and an interconnected tunnel‐like pyrochlore crystal framework, is a promising high‐capacity energy‐storage material. Fabricating electrochemically reversible PAA electrode materials for alkali‐metal‐ion storage is a challenge and has never been reported due to the extremely poor intrinsic electronic conductivity of PAA associated with the highest oxidation state Sb(V). Combining nanostructure engineering with a conductive‐network construction strategy, here is reported a facile one‐pot synthesis protocol for crafting uniform internal‐void‐containing PAA nano‐octahedra in a composite with nitrogen‐doped reduced graphene oxide nanosheets (PAA?N‐RGO), and for the first time, realizing the reversible storage of both Li⁺ and K⁺ ions in PAA?N‐RGO. Such an architecture, as validated by theoretical calculations and ex/in situ experiments, not only fully takes advantage of the large‐sized tunnel transport pathways (0.37 nm²) of PAA for fast solid‐phase ionic diffusion but also leads to exponentially increased electrical conductivity (3.3 S cm^?1 in PAA?N‐RGO vs 4.8 × 10^?10 S cm^?1 in bare‐PAA) and yields an inside‐out buffer function for accommodating volume expansion. Compared to electrochemically irreversible bare‐PAA, PAA?N‐RGO manifests reversible conversion‐alloying of Sb(V) toward fast and durable Li⁺‐ and K⁺‐ion storage.     

Intrinsic curvature in normal and inverted lipid structures and in membranes.

The intrinsic or spontaneous radius of curvature, R(o), of lipid monolayer assemblies is expressed in terms of a lipid molecular packing parameter, V/AI, for various geometries. It is shown that the equivalent lipid length, 1, in inverted hexagonal (HII) phases, defined by a cylindrical shell of equal total lipid volume, yields an expression for R o identical to that for inverted cylindrical micelles (or, equivalently, HII phases in the presence of excess hydrocarbon). This identity is used to obtain values of the effective packing parameter for various phosphatidylethanolamines. The temperature dependence of the intrinsic radius of curvature is predicted to be negative and to be considerably greater than that for the lipid length in nearly all cases. The thermal expansion coefficient is not constant but is found to vary, depending on the value of the lipid packing parameter. A possible addition rule is constructed for the intrinsic radius of curvature of lipid mixtures, based on the linear additivity of the effective molecular volumes, V, and molecular areas, A. This relation is found to hold for mixtures of dioleoyl phosphatidylcholine (DOPC) with dioleoyl phosphatidylethanolamine, and a value of R(o) of > or = 9 A (V/AI = 1.08) is obtained for DOPC. The energetics of the intrinsic curvature and lamellar-nonlamellar transitions are also discussed within the framework of the model.     

Shell features, main colonized environments, and fractal analysis of sutures in Late Jurassic ammonites

A precise approach to the quantification of relationships between suture complexity, as measured by fractal analysis (step-line procedure), the architecture of shells, and the main colonized environments, has been made in a set of Late Jurassic ammonites ( N =507). Statistically significant differences between fractal-dimension ( D _f) mean values of evolute and involute shells are interpreted as caused by differences in the surface:volume ( S:V ) ratio. Suture complexity is also related to the shape of whorl section. The lowest D _f values correspond to subcircular whorl sections (low S:V ratio) and the highest ones to acute sections (high S:V ratio). The shape of flanks shows correlation with suture complexity. The highest values of D _f are found in planulate shells and the lowest ones in whorl cross sections with convex flanks. Highly significant differences appear between D _f mean values from unsculptured shells and those from ammonites with ribs and/or tubercles of medium to large size. Multivariate analysis shows a combined variation of shell features and suture complexity, resulting in a heterogeneous distribution of D _f values within the ammonite morphospace, mainly according to structural (shell architecture) and ornamental (sculpture strength rather than density) factors. Finally, the data obtained on relationships between suture complexity and the colonized environments (epicontinental vs. epioceanic inhabitants) suggest that suture complexity is not primarily related to bathymetry, and/or that no major differences in habitat depths existed between epicontinental and epioceanic ammonites.     

Storage versus substrate limitation to bole respiratory potential in two coniferous tree species of contrasting sapwood width

Two coniferous tree species of contrasting sapwood width (Pinus ponderosa L., ponderosa pine and Pseudotsuga menziesii Mirb., Douglas-fir) were compared to determine whether bole respiratory potential was correlated with available storage space in ray parenchyma cells and/or respiratory substrate concentration of tissues (total nitrogen content, N; and total non-structural carbohydrate content, TNC). An increment core-based, laboratory method under controlled temperature was used to measure tissue-level respiration (termed respiratory potential) from multiple positions in mature boles (>100-years-old). The most significant tissue-level differences that occurred were that N and TNC were two to six times higher for inner bark than sapwood, TNC was about two times higher in ponderosa pine than Douglas-fir and there was significant seasonal variation in TNC. Ray cell abundance was not correlated with sapwood respiratory potential, whereas N and TNC often were, implying that respiratory potential tended to be more limited by substrate than storage space. When scaled from cores to whole boles (excluding branches), potential net CO2 efflux correlated positively with live bole volume (inner bark plus sapwood), live bole ray volume, N mass, and TNC mass (adjusted R2 > or =0.4). This relationship did not differ between species for N mass, but did for live bole volume, live bole ray volume, and TNC mass. Therefore, N mass appeared to be a good predictor of bole respiratory potential. The differences in net CO2 efflux between the species were largely explained by the species' relative amounts of whole-bole storage space or substrate mass. For example, ponderosa pine's inner bark was thinner than Douglas-fir's, which had the greater concentration of ray cells and TNC compared with the sapwood. This resulted in ponderosa pine boles having 30-60% less ray volume and 10-30% less TNC mass, and caused ponderosa pine net CO2 efflux/ray volume and net CO2 efflux/TNC mass to be 20-50% higher than Douglas-fir. In addition, because inner bark respiratory potential was 2-25 times higher than that of sapwood, ponderosa pine's thinner inner bark and deeper sapwood (relative to Douglas-fir) caused its bole net CO2 efflux/live bole volume to be 20-25% lower than that of similarly-sized Douglas-fir trees.     

Cavitation and water storage capacity in bole xylem segments of mature and young Douglas-fir trees

Hydraulic specific conductivity, vulnerability to cavitation and water storage capacity of Douglas-fir sapwood was determined for samples from six young (1.0-1.5 m tall) and six mature trees (41-45 m tall). Measurements on samples from young trees showedthere were no effects of two contrasting sample types (entire stem segments vs sectors chiseled out of entire stems) on any of the calculated hydraulic parameters, for vulnerability to cavitation and water storage capacity. Measurements on mature trees were made on wood from four heights on the bole and from two sapwood depths. Outer and inner sapwood at the base of the tree had higher water storage capacities and were more vulnerable to cavitation than was sapwood from the tree top. At every height, old trees were more vulnerable to cavitation than at 1.0 m from the ground in young trees. The water storage capacities showed three distinct phases at the base of the trunk. Young trees had similar water storage capacity (per unit volume of sapwood) to the topof the mature trees, which was lower than the water storage capacity throughout the rest of the bole xylem. The way in which capacitance was calculated (on a volumetric basis vs a relative water content basis) affected the conclusion one would draw at the low water potentials (<-3 MPa). Within a tree, we found an apparent trade-off between having both hydraulic specific conductivity and stem water storage, and vulnerability to cavitation.     

Bacteriophage phiNS11: a lipid-containing phage of acidophilic thermophilic bacteria. IV. Sedimentation coefficient, diffusion coefficient, partial specific volume, and particle weight of the phage.

The particle weight (molecular weight) of phiNS11 was determined from the sedimentation coefficient, diffusion coefficient, and partial specific volume of the phage. The sedimentation coefficient of the phage (S(0)20, W) is 416 +/- 2.7S. The diffusion coefficient D(0)20, W), which was determined by quasielastic light scattering measurement, is (0.57 +/- 0.03) x 10(-7) cm2/s. The partial specific volume was determined by the mechanical oscillation technique to be 0.747 +/- 0.007 cm3/g. Based on these values, the particle weight of the phage was calculated to be (70.3 +/- 4.3) x 10(6) daltons, which agrees well with the particle weight (69--72 x 10(6) daltons) estimated from the molecular weight of phage DNA and the content of DNA. The Stokes radius of the phage particle was calculated to be 37.7 +/- 2 nm and hydration of the phage was estimated to be 1.18 cm3/g of dry phage. From the particle weight and the chemical composition of the phage, we estimated that one phage particle contains one double-stranded DNA molecule, 16,000 residues of fatty acid, 72 protein I molecules, 920 protein II, 42 protein III, 48 protein IV, 290 protein V molecules, and 3,700 molecules of polyamines.     

Influence of aspect ratios on the creep behaviour of articular cartilage in indentation

Indentation tests are commonly used to determine the mechanical behaviour of articular cartilage with varying properties, thickness, and geometry. This investigation evaluated the effect of changing geometric parameters on the properties determined from creep indentation tests. Finite element analyses simulated the indentation behaviour of two models, an excised cylindrical specimen of cartilage with either normal and repair qualities and an osteochondral defect represented as a cylindrical region of repair cartilage integrated with a surrounding layer of normal tissue. For each model, the ratios of indenter radius to cartilage height (a/h=0.5,1.5) and cartilage radius to indenter radius (r/a=2,5) were varied. The vertical displacement of the cartilage under the indenter obtained through finite element analysis was fitted to a numerical algorithm to determine the aggregate modulus, permeability, and Poisson's ratio. Indentation behaviours of cartilage specimens for either model with a/h=1.5 were not affected by r/a for values of 2 and 5. Aggregate modulus was not greatly affected by the geometric changes studied. Permeability was affected by changes in the ratio of specimen to indenter radii for a/h=0.5. These findings suggest that experimental configurations of excised cylindrical specimens, also representing osteochondral defects with no or unknown degree of integration, where the cartilage layer has a/h=0.5 should not have r/a values on the order of 2 for confidence in the mechanical properties determined. Indentation of osteochondral defects where the repair cartilage is fully integrated to the surrounding cartilage can be performed with confidence for all cases tested.     

Fire and drought: Shifts in bark investment across a broad geographical scale for Neotropical savanna trees

Savanna tree communities occurring in confluence zones with other biomes likely experience different environmental pressures, resulting in shifts in the selection of individual traits, the combinations of such traits, and species composition. In seasonally dry fire-prone environments, plant survival is presumably associated with adaptive changes in bark properties related to fire protection and water storage. Here, we integrated the multiple functions of the bark to investigate whether different selective pressures could influence patterns of variation in bark structure and allocation across species in a broad geographical range. We measured thickness, density, and water content of the inner and outer bark in branches and the main stem of the 51 most abundant species in three savanna communities differing in climatic aridity, one located at the core region of Cerrado in Central Brazil and the other two at its periphery, in the transition zones with Amazonia and Atlantic forest biomes. We found no difference in outer bark thickness but markedly difference in inner bark thickness between the three plant communities. In the central region, where dry season is long and fire is frequent, branches and main stem showed thicker inner bark. Contrastingly, in the south periphery region, where dry season is short, species showed thinner inner bark in both branches and main stem. Species from the north periphery region, where mean annual precipitation is higher, but fire is frequent and the dry season is also long, showed similar main stem inner bark thickness, but thinner branch inner bark compared to core region species. Our findings support the idea that investing in inner bark thickness and bark moisture may be the most advantageous strategy in plant communities that suffer from high evaporative demand during a long period and are at a high risk of fire.     

Fire severity and nutrient availability do not constrain resprouting in forest shrubs

Plants often survive disturbances such as fire by resprouting which involves having both protection traits and carbohydrate storage capacity. Protection traits not only act directly to insulate meristems but also prevent combustion of carbohydrate stores. Rapid stem growth also allows replenishment of carbohydrate stores ensuring persistence through another event. Resource availability may, however, constrain the ability to develop resilience to high-severity fires through either nutrient limitation or light limitation. We tested whether fire severity influenced resprouting ability of woody plants in two contrasting environments, low nutrient dry sclerophyll forest and more fertile wet sclerophyll forest. We tested which fire protection and growth traits were associated with resprouting ability (27 species) and resprouting vigour (16 species). Fire severity did not limit the ability of most species to resprout in either forest type. There was no generalized protection syndrome for surviving top kill, but combinations of bud protection and growth together with storage capacity appear to drive resprouting ability. In nutrient-limited forests, low specific leaf area (SLA) may reduce stem growth in resprouters, causing more reliance on bud protection through bark thickness. Conversely, in the more fertile forests, where light becomes limiting with time-since-fire, high SLA appears to increase the capacity for rapid stem growth with less emphasis on developing thicker bark. These different syndromes appear to be adaptive as fire severity did not influence survival in either forest type.     

Molecular dynamics and mutational analysis of a channelopathy mutation in the IIS6 helix of Ca V 1.2

A channelopathy mutation in segment IIS6 of Ca(V)1.4 (I745T) has been shown to cause severe visual impairment by shifting the activation and inactivation curves to more hyperpolarized voltages and slowing activation and inactivation kinetics. A similar gating phenotype is caused by the corresponding mutation, I781T, in Ca(V)1.2 (midpoint of activation curve (V(0.5)) shifted to -37.7 +/- 1.2 mV). We show here that wild-type gating can partially be restored by a helix stabilizing rescue mutation N785A. V(0.5) of I781T/N785A (V(0.5) = -21.5 +/- 0.6 mV) was shifted back towards wild-type (V(0.5) = -9.9 +/- 1.1 mV). Homology models developed in our group (see accompanying article for details) were used to perform Molecular Dynamics-simulations (MD-simulations) on wild-type and mutant channels. Systematic changes in segment IIIS6 (M1187-F1194) and in helix IIS6 (N785-L786) were studied. The simulated structural changes in S6 segments of I781T/N785A were less pronounced than in I781T. A delicate balance between helix flexibility and stability enabling the formation of hydrophobic seals at the inner channel mouth appears to be important for wild-type Ca(V)1.2 gating. Our study illustrates that effects of mutations in the lower part of IIS6 may not be localized to the residue or even segment being mutated, but may affect conformations of interacting segments.     

Subzero water transport characteristics of boar spermatozoa confirm observed optimal cooling rates

Incomplete understanding of the water transport parameters (reference membrane permeability, L(pg), and activation energy, E(Lp)) during freezing in the presence of extracellular ice and cryoprotective agents (CPAs) is one of the main limiting factors in reconciling the difference between the numerically predicted value and the experimentally determined optimal rates of freezing in boar (and in general mammalian) gametes. In the present study, a shape-independent differential scanning calorimeter (DSC) technique was used to measure the water transport during freezing of boar spermatozoa. Water transport data during freezing of boar sperm cell suspensions were obtained at cooling rates of 5 and 20 degrees C/min in the presence of extracellular ice and 6% (v/v) glycerol. Using previously published values, the boar sperm cell was modeled as a cylinder of length 80.1 microm and a radius of 0.31 microm with an osmotically inactive cell volume, V(b), of 0.6 V(o), where V(o) is the isotonic cell volume. By fitting a model of water transport to the experimentally obtained data, the best-fit water transport parameters (L(pg) and E(Lp)) were determined. The "combined-best-fit" parameters at 5 and 20 degrees C/min for boar spermatozoa in the presence of extracellular ice are: L(pg) = 3.6 x 10(-15) m(3)/N. s (0.02 microm/min-atm) and E(Lp) = 122.5 kJ/mole (29.3 kcal/mole) (R(2) = 0.99); and the corresponding parameters in the presence of extracellular ice and glycerol are: L(pg)[cpa] = 0.90 x 10(-15) m(3)/N. s (0.005 microm/min-atm) and E(Lp)[cpa] = 75.7 kJ/mole (18.1 kcal/mole) (R(2) = 0.99). The water transport parameters obtained in the present study are significantly different from previously published parameters for boar and other mammalian spermatozoa obtained at suprazero temperatures and at subzero temperatures in the absence of extracellular ice. The theoretically predicted optimal rates of freezing using the new parameters ( approximately 30 degrees C/min) are in close agreement with previously published but experimentally determined optimal cooling rates. This analysis reconciles a long-standing difference between theoretically predicted and experimentally determined optimal cooling rates for boar spermatozoa.     

In vivo DNA protection by relaxed-specificity SinI DNA methyltransferase variants

The SinI DNA methyltransferase, a component of the SinI restriction-modification system, recognizes the sequence GG(A/T)CC and methylates the inner cytosine to produce 5-methylcytosine. Previously isolated relaxed-specificity mutants of the enzyme also methylate, at a lower rate, GG(G/C)CC sites. In this work we tested the capacity of the mutant enzymes to function in vivo as the counterpart of a restriction endonuclease, which can cleave either site. The viability of Escherichia coli cells carrying recombinant plasmids with the mutant methyltransferase genes and expressing the GGNCC-specific Sau96I restriction endonuclease from a compatible plasmid was investigated. The sau96IR gene on the latter plasmid was transcribed from the araBAD promoter, allowing tightly controlled expression of the endonuclease. In the presence of low concentrations of the inducer arabinose, cells synthesizing the N172S or the V173L mutant enzyme displayed increased plating efficiency relative to cells producing the wild-type methyltransferase, indicating enhanced protection of the cell DNA against the Sau96I endonuclease. Nevertheless, this protection was not sufficient to support long-term survival in the presence of the inducer, which is consistent with incomplete methylation of GG(G/C)CC sites in plasmid DNA purified from the N172S and V173L mutants. Elevated DNA ligase activity was shown to further increase viability of cells producing the V173L variant and Sau96I endonuclease.     

Morphological changes along an altitude gradient and their consequences for an andean giant rosette plant

Summary Selected morphological features were measured in five populations of the giant rosette plant Espeletia schultzii occurring along an elevation gradient from 2600 to 4200 m in the Venezuelan Andes. Pith volume per amount of leaf area increases with elevation resulting in significantly larger water storage capacity at higher elevations. Thickness of leaf pubescence and, therefore, leaf boundary layer resistance, also increases with elevation resulting in both potentially higher leaf temperatures relative to air temperature and higher leaf to air vapor pressure gradients. The net effect on transpiration rate would depend on ratios of stomatal to boundary layer resistance and leaf energy balance. At higher elevations the central rosette leaves are more vertically oriented and the leaf bases show a pronounced curvature as the intersection with the main axis is approached. This gives these rosettes a distinctly paraboloid appearance and probably enhances capture and retention of incident long and shortwave radiation by the apical bud and expanding leaves. Features which result in enhanced water storage capacity and higher plant temperatures relative to air temperature without greatly increasing water loss are adaptive in high altitude paramo habitats where water availability and growth are limited by year round low temperatures (mean 2–3° C).     

Model of gas transport during high-frequency ventilation

We analyze gas exchange during high-frequency ventilation (HFV) by a stochastic model that divides the dead space into N compartments in series where each compartment has a volume equal to tidal volume (V). We then divide each of these compartments into alpha subcompartments in series, where each subcompartment receives a well-mixed concentration from one compartment and passes a well-mixed concentration to another in the direction of flow. The number of subcompartments is chosen on the basis that 1/alpha = (sigma t/-t)2, where -t is mean transit time across a compartment of volume, and sigma t is standard deviation of transit times. If (sigma t/-t)D applies to the transit times of the entire dead space, the magnitude of gas exchange is proportional to (sigma t/-t)D, frequency, and V raised to some power greater than unity in the range where V is close to VD. When V is very small in relation to VD, gas exchange is proportional to (sigma t/-t)2D, frequency, and V raised to a power equal to either one or two depending on whether the flow is turbulent or streamline, respectively. (sigma t/-t)D can be determined by the relation between the concentration of alveolar gas at the air outlet and volume expired as in a Fowler measurement of the volume of the dead space.