Stability of the replicator equation for a single species with a multi-dimensional continuous trait space

The replicator equation model for the evolution of individual behaviors in a single species with a multi-dimensional continuous trait space is developed as a dynamics on the set of probability measures. Stability of monomorphisms in this model using the weak topology is compared to more traditional methods of adaptive dynamics. For quadratic fitness functions and initial normal trait distributions, it is shown that the multi-dimensional continuously stable strategy (CSS) of adaptive dynamics is often relevant for predicting stability of the measure-theoretic model but may be too strong in general. For general fitness functions and trait distributions, the CSS is related to dominance solvability which can be used to characterize local stability for a large class of trait distributions that have no gaps in their supports whereas the stronger neighborhood invader strategy (NIS) concept is needed if the supports are arbitrary.     

Evolutionary and dynamic stability in continuous population games

 Asymptotic stability under the replicator dynamics over a continuum of pure strategies is shown to crucially depend on the choice of topology over the space of mixed population strategies, namely probability measures over the real line. Thus, Strong Uninvadability, proved by Bomze (1990) to be a sufficient condition for asymptotic stability under the topology of variational distance between probability measures, implies convergence to fixation over a pure strategy x ^* only when starting from a population strategy which assigns to x ^* a probability sufficiently close to one. It does not imply convergence to x ^* when starting from a distribution of small deviations from x ^* , regardless of how small these deviations are. It is, therefore, suggested that when a metric space of pure strategies is involved, another topology, hence another stability condition, may prove more relevant to the process of natural selection. Concentrating on the case of a one dimensional continuous quantitative trait, we resort to the natural Maximum Shift Topology in which an ɛ-vicinity of the fixation on a pure strategy x ^* consists of all mixed population strategies with support which includes x ^* and is in the ɛ-neighborhood of x ^* . Under this topology, a relatively simple necessary and sufficient condition for replicator asymptotic stability, namely Continuous Replicator Stability (CRSS), is demonstrated. This condition is closely related to the static stability condition of Neighbor Invadability (Apaloo 1997), and slightly stronger than the condition of Continuous Stability (Eshel and Motro 1981). Received: 9 July 2002 / Revised version: 12 November 2002 / Published online: 19 March 2003 Key words or phrases: CSS – ESS – NIS – Strong uninvadability – Weak topology – Replicator dynamics – Long-term evolution – Continuous population games – Continuous replicator stability – Asymptotic stability     

Single species evolutionary dynamics

Not long after the introduction of evolutionary stable strategy (ESS) concept, it was noticed that dynamic selection did not always lead to the establishment of the ESS. The concept of continuously stable strategy (CSS) was thereafter developed. It was generally accepted that dynamic selection leads to the establishment of an ESS if it is a CSS. Examination of an evolutionary stability concept which is called neighborhood invader strategy (NIS) shows that it may be impossible for an ESS to be established through dynamic selection even if it is a CSS and no polymorphisms occur. We will examine the NIS concept and its implications for two evolutionary game models: root-shoot allocation in plant competition and Lotka–Volterra competition. In the root-shoot model we show that an ESS will be attained through dynamic selection if it is a NIS. Similarly for the Lotka–Volterra model, we show that an ESS will be attained through dynamic selection even if protected dimorphisms occur during the evolutionary process if it is an NIS.     

Multi-species evolutionary dynamics

Dynamical attainability of an evolutionarily stable strategy (ESS) through the process of mutations and natural selection has mostly been addressed through the use of the continuously stable strategy (CSS) concept for species evolutionary games in which strategies are drawn from a continuum, and by the adaptive trait dynamics method. We address the issue of dynamical attainability of an ESS in coevolving species through the use of the concept of an ESNIS. It is shown that the definition of an ESNIS coalition for coevolving species is not in general equivalent to other definitions for CSS given in the literature. We show under some additional conditions that, in a dynamic system which involves the strategies of a dimorphic ESNIS coalition and at most two strategies that are not members of ESNIS coalition, the ESNIS coalition will emerge as the winner. In addition an ESNIS will be approached because of the invasion structure of strategies in its neighborhood. This proves that under the above conditions an ESNIS has a better chance of being attained than a strategy coalition which is a CSS. The theory developed is applied to a class of coevolutionary game models with Lotka–Volterra type interactions and we show that for such models, an ESS coalition will be dynamically attainable through mutations and natural selection if the ESS coalition is also an ESNIS coalition.Co-ordinating editor: Metz     

Na+/I− Symporter and Type 3 Iodothyronine Deiodinase Gene Expression in Amniotic Membrane and Placenta and Its Relationship to Maternal Thyroid Hormones

Placental type 3 iodothyronine deiodinase (D3) potentially protects the fetus from the elevated maternal thyroid hormones. Na⁺/I^? symporter (NIS) is a plasma membrane glycoprotein, which mediates active iodide uptake. Our objectives were to establish the distribution of NIS and D3 gene expressions in the placenta and the amniotic membrane and to investigate the relationship between placental D3 and NIS gene expressions and maternal iodine, selenium, and thyroid hormone status. Thyroid hormones, urinary iodine concentration (UIC), and selenium levels were measured in 49 healthy term pregnant women. NIS and D3 gene expressions were studied with the total mRNA RT-PCR method in tissues from maternal placenta (n?=?49), fetal placenta (n?=?9), and amniotic membrane (n?=?9). NIS and D3 gene expressions were shown in the fetal and maternal sides of the placenta and amniotic membrane. Mean blood selenium level was 66?±?26.5 μg/l, and median UIC was 143 μg/l. We could not demonstrate any statistically significant relationship of spot UIC and blood selenium with NIS and D3 expression (p?>?0.05). Positive correlations were found between NIS and thyroxine-binding globulin (TBG) (r?=?0.3, p?=?0.042) and between D3 and preoperative glucose levels (r?=?0.4, p?=?0.006). D3 and NIS genes are expressed in term placenta and amniotic membrane; thus, in addition to placenta, amniotic membrane contributes to regulation of maternofetal iodine and thyroid hormone transmission. Further studies are needed to clarify the relationship between maternal glucose levels and placental D3 expression and between TBG and placental NIS expression.     

Evolutionarily stable strategy and invader strategy in matrix games

In this paper, we consider the concepts of evolutionarily stable strategy (ESS), neighborhood invader strategy (NIS) and global invader strategy (GIS) in single species with frequency-dependent interactions. We find some general relationships among the three concepts in matrix games. The main conclusion is that ESS and NIS are equivalent to each other and are both equivalent to local superiority; a strategy with global superiority must be a GIS; a GIS may not be equivalent to its global superiority in games with more than two players; and in any two-player matrix game a GIS is just equivalent to its global superiority. In two-player games, globally asymptotic stability in the replicator dynamics has also been shown. Equivalent conditions for the three concepts stated by payoff comparisons are given and are applied to examples involved.     

养分资源脉冲供给对几种微藻种间竞争的影响

以淡水生态系统常见的5种微藻为研究对象,通过稳态条件下单一培养的方式获取各微藻在氮素或磷素缺乏条件下对应的生长特征参数和R*值,同时将5种微藻在养分资源脉冲供给的方式下进行混合培养,进而检测养分资源脉冲供给对几种微藻种间竞争的影响作用,并与基于稳态条件下所预测的微藻种间竞争结果进行比较。研究结果显示,在稳态条件下具有最小R*值的纤细角星鼓藻在与其它微藻的种间竞争过程中始终保持优势地位,而其余4种微藻在混合培养状态下的相对比重亦与其稳态条件下的R*值紧密相关。然而,资源脉冲供给条件下的竞争优势种纤细角星鼓藻能与其它种群数量较小的微藻共存。此外,在氮素和磷素的不同供给比例情况下,对应微藻群落的结构也会发生相应的变化。实验养分资源脉冲作用下多种微藻的共存现象与自然水体中的观测现象相一致,显示了资源脉冲可能是维持生物多样性水平的一个重要机制。     

A mixed strategy model for the emergence and intensification of social learning in a periodically changing natural environment

Based on a population genetic model of mixed strategies determined by alleles of small effect, we derive conditions for the evolution of social learning in an infinite-state environment that changes periodically over time. Each mixed strategy is defined by the probabilities that an organism will commit itself to individual learning, social learning, or innate behavior. We identify the convergent stable strategies (CSS) by a numerical adaptive dynamics method and then check the evolutionary stability (ESS) of these strategies. A strategy that is simultaneously a CSS and an ESS is called an attractive ESS (AESS). For certain parameter sets, a bifurcation diagram shows that the pure individual learning strategy is the unique AESS for short periods of environmental change, a mixed learning strategy is the unique AESS for intermediate periods, and a mixed learning strategy (with a relatively large social learning component) and the pure innate strategy are both AESS's for long periods. This result entails that, once social learning emerges during a transient era of intermediate environmental periodicity, a subsequent elongation of the period may result in the intensification of social learning, rather than a return to innate behavior.     

Diurnal changes in volume and solute transport coefficients of phaseolus roots

Volume (J_v) and solute (J_s) fluxes through Phaseolus root systems were observed over a 24-hour period. The volume flux was varied in a pressure chamber by altering the hydrostatic pressure in 10 steps, from 0 to 0.41 megapascals. All root systems showed strong diurnal peaks in volume flux. The five transport coefficients (σ, ω, J_s^*, L_p, and π^*) were estimated from a nonlinear least squares algorithm. Analysis of the data revealed that all the coefficients exhibited a diurnal rhythm. When the total differential of the volume flux was considered it was possible to show that the diurnal changes in volume flux were due to a complex interaction between the diurnally shifting coefficients with the role of each highly dependent on the level of volume flux. At low volume fluxes, ω, J_s^*, and π^* accounted for nearly all the diurnal change in volume flux. At high volume fluxes, however, the major influence shifted to L_p and π^*, while ω and J_s^* became relatively unimportant. Thus, π^* was the only coefficient of interest across the entire range of J_v and appeared to be the single most important one in determining the diurnal rhythm of J_v under conditions of a constant applied pressure.     

The utility and performance of near-infra red spectroscopy in simultaneous monitoring of multiple components in a high cell density recombinant Escherichiacoli production process

Bioprocess optimisation is often limited by an inability to measure biomass, nutrient and by-product concentrations in a time frame which allows process adjustments. Near-infrared (near-IR) spectroscopy can potentially be used to measure each of these components within 2 minutes of sampling, using an unprocessed whole broth sample. In the present study the use of near-IR spectroscopy for at-line (rapid off-line) monitoring of biomass, glycerol, ammonium, and acetate in a recombinant Escherichia coli fed batch process was investigated. The following robust correlation models were developed for these analytes using multiple least squares linear regression (MLR): [Glycerol],gl^?1 =15.957 ? 2219.270^* A(λ₂₂₇₄)?1705.041^* A(λ₂₁₇₂); [Acetate],gl^?1=27.683 ?1757.258^* A(λ₂₂₅₄)+296.903^* A(λ₂₃₄₀)?21.325^* A (λ₆₂₀); [Ammonium],gl^?1=?1310.502?47912.960^* A (λ₂₁₄₈)?135149.300^* A(λ₁₇₈₂)?27636.200^* A (λ₈₃₀); and [Biomass],gl^?1=14.034?3.548^* A(λ₆₀₂/λ₁₁₃₄) ?4286.050^* A(λ₉₂₈). Using these models permitted rapid simultaneous analysis of all four analytes. This improved monitoring capability was used to develop a high cell density recombinant E. coli fed-batch process in which ammonium and acetate accumulation were minimised leading to higher cell densities. By manipulation of the C?:?N ratio in the complex feed, the toxic effects of ammonium accumulation upon the organism were minimised, thereby facilitating the application of a carbon limited feeding strategy. The effect of these C?:?N ratio medium changes, upon the near-IR measurement capability, was investigated. In this process, near-IR spectroscopy has been shown to be a powerful, accurate and precise method for simultaneously measuring several key process variables. Its accuracy, precision and utility for at-line measurement and control are evaluated, particularly in reference to processes where the initial medium composition may vary, leading to changes in the chemical matrix. The potential of near-infra red for online analysis and control is discussed.     

Capillary operators

The goal of this work is an examination of capillary exchange models as mathematical operators. The concentration function relations for the Krogh cylinder of a single capillary, basic to many organ models, are studied via the theory of operators on the Lebesgue normed spacesL _p[0,∞], (1<-p<-∞). A discussion is included of theL _p -normsvis-à-vis the coefficient of variation currently used in finding capillary parameters and evaluating parameter searches. The capillary model determines two operators on the space of locally integrable functions: O _K (relating extravascular concentration to intravascular) and K _{a, k} (relating intravascular concentration to input), wherek is the ratio of permeabilitysurface area (PS) to extravascular volume, and α is the ratio of PS to flow. These operators are shown to induce contractive (‖O _K ‖ _p <-1, ‖K _{a, k} ‖ _p <-1), isotone, linear operators onL _p . The uniform convergence relation $$K_{a,k} = \mathop {\lim _{(p)} }\limits_{N \to \infty } \left( {\sum\limits_{n = 0}^N {P_n (a)O_k^n } } \right)$$ (as operators onL _p) is derived, whereP _n (a) is the Poisson probabilitye ^?a a ⁿ /n!. For the important special cases ofp=∞, 1, 2 the norms are found (‖O_k‖=‖K_a,k‖_p=1). Consideration is also given to the norms and operators when the functions involved are limited to a finite interval of time.     

In vivo Molecular Imaging and Radionuclide (131I) Therapy of Human Nasopharyngeal Carcinoma Cells Transfected with a Lentivirus Expressing Sodium Iodide Symporter

Introduction

Despite recent improvements in the survival rates for nasopharyngeal carcinoma (NPC), novel treatment strategies are required to improve distant metastasis-free survival. The sodium iodine symporter (NIS) gene has been applied for in vivo imaging and cancer therapy. In this study, we examined the potential of NIS gene therapy as a therapeutic approach in NPC by performing non-invasive imaging using ¹²⁵I and ¹³¹I therapy in vivo.

Methods

We constructed a lentiviral vector expressing NIS and enhanced green fluorescent protein (EGFP) under the control of the human elongation factor-1α (EF1α) promoter, and stably transfected the vector into CNE-2Z NPC cells to create CNE-2Z-NIS cells. CNE-2Z and CNE-2Z-NIS tumor xenografts were established in nude mice; ¹²⁵I uptake, accumulation and efflux were measured using micro-SPECT/CT imaging; the therapeutic effects of treatment with ¹³¹I were assessed over 25 days by measuring tumor volume and immunohistochemical staining of the excised tumors.

Results

qPCR, immunofluorescence and Western blotting confirmed that CNE-2Z-NIS cells expressed high levels of NIS mRNA and protein. CNE-2Z-NIS cells and xenografts took up and accumulated significantly more ¹²⁵I than CNE-2Z cells and xenografts. In vitro, ¹³¹I significantly reduced the clonogenic survival of CNE-2Z-NIS cells. In vivo, ¹³¹I effectively inhibited the growth of CNE-2Z-NIS xenografts. At the end of ¹³¹I therapy, CNE-2Z-NIS xenograft tumor cells expressed higher levels of NIS and caspase-3 and lower levels of Ki-67.

Conclusion

Lentiviruses effectively delivered and mediated long-lasting expression of NIS in CNE-2Z cells which enabled uptake and accumulation of radioisotopes and provided a significant therapeutic effect in an in vivo model of NPC. NIS-mediated radioiodine treatment merits further investigation as a potentially effective, low toxicity therapeutic strategy for NPC.     

On a possible biological spectroscopy

A new type of physical transition, denotedS→S ^*, has been detected in irradiated organic molecules (λ=546 nm) through their interaction with specific biological macromolecules. In a specific enzyme-substrate interaction, a clear enhancement of the reaction rate is observed, when the substrate is irradiated with sharply well defined times. These “efficient irradiation times” are always of the 5k sec type (k=1, 2, 3, …). They have been consistently revealed in a great number of specific biological interactions. The present note demonstrates an important property, i.e. that forevery irradiation time aS→S ^* transition is induced in organic molecules. It is shown that for any irradiation times different from the 5k sec type (k=1, 2, 3, …) states of theS ^* type may occur, but the biological macromolecules may “detect” only theS ^* states induced by irradiations of the 5k sec type.     

Beyond non-integer Hill coefficients: A novel approach to analyzing binding data,applied to Na+-driven transporters

Prokaryotic and eukaryotic Na⁺-driven transporters couple the movement of one or more Na⁺ ions down their electrochemical gradient to the active transport of a variety of solutes. When more than one Na⁺ is involved, Na⁺-binding data are usually analyzed using the Hill equation with a non-integer exponent n. The results of this analysis are an overall K_d-like constant equal to the concentration of ligand that produces half saturation and n, a measure of cooperativity. This information is usually insufficient to provide the basis for mechanistic models. In the case of transport using two Na⁺ ions, an n < 2 indicates that molecules with only one of the two sites occupied are present at low saturation. Here, we propose a new way of analyzing Na⁺-binding data for the case of two Na⁺ ions that, by taking into account binding to individual sites, provides far more information than can be obtained by using the Hill equation with a non-integer coefficient: it yields pairs of possible values for the Na⁺ affinities of the individual sites that can only vary within narrowly bounded ranges. To illustrate the advantages of the method, we present experimental scintillation proximity assay (SPA) data on binding of Na⁺ to the Na⁺/I⁻ symporter (NIS). SPA is a method widely used to study the binding of Na⁺ to Na⁺-driven transporters. NIS is the key plasma membrane protein that mediates active I⁻ transport in the thyroid gland, the first step in the biosynthesis of the thyroid hormones, of which iodine is an essential constituent. NIS activity is electrogenic, with a 2:1 Na⁺/I⁻ transport stoichiometry. The formalism proposed here is general and can be used to analyze data on other proteins with two binding sites for the same substrate.     

Properties of a general model of DNA evolution under no-strand-bias conditions

Under the hypothesis of no-strand-bias conditions, the Watson and Crick base-pairing rule decreases the complexity of models of DNA evolution by reducing to six the maximum number of substitution rates. It was shown that intrastrand equimolarity between A and T (A ^* T ^*) and between G and C (G ^* C ^*) is a general asymptotic property of this class of models. This statistical prediction was observed on 60 long genomic fragments (>50 kbp) from various kingdoms, even when the effect of the two opposite orientations for coding sequences is removed. The practical consequence of the model for estimating the expected number of substitutions per site between two homologous DNA sequences is discussed.Abbreviations BPR Watson and Crick base pairing rule (A:T, G:C) - PRI Intrastrand type-1 parity rule (i j, m(i,j)m( )) - PRII Intra strand type-2 parity rule (A ^* T ^*, G ^* C ^*)     

Genetic Population Structure of Pacific Hake, Merluccius productus, in the Pacific Northwest

This study presents the first protein electrophoretic study of population structure within the Georgia Basin Pacific hake Distinct Population Segment, as defined under the U.S. Endangered Species Act. Forty-one allozyme loci (29 polymorphic) were analyzed in samples from three Pacific hake spawning populations on the west coast of North America: (1) Port Susan, Puget Sound, Washington (three temporal samples); (2) south-central Strait of Georgia, British Columbia, Canada (two temporal samples); and (3) offshore of southern California (two temporal samples) (total n = 664). Mean heterozygosity over all loci was 12–13% for all populations. Within-population temporal samples were not significantly different from one another, but statistically significant differences were detected at 15 of the 29 polymorphic loci (p < 0.05) among the three populations. Differences at eight of these loci were highly significant (p < 0.001): ADA ^*, ALAT ^*, bGALA ^*, GPI-A ^*, sIDHP ^*, LDH-A ^*, MPI ^*, and PEP-B ^*. The two Georgia Basin populations were significantly different at six loci: bGALA ^*, sIDHP ^*, LDH-A ^*, MPI ^*, PGK ^*, and PGM-2 ^* (p < 0.05). Nei's genetic distance (D) was 0.0006 between Port Susan and Strait of Georgia pooled temporal samples, and 0.005 between these populations and offshore Pacific hake. F_ST was 0.02 and 0.0046 among all three populations and among the Georgia Basin populations, respectively. Both F_ST estimates were significantly greater than zero, and the results suggest a high degree of demographic isolation among all three populations.     

A New Screening Method for the Directed Evolution of Thermostable Bacteriolytic Enzymes

Directed evolution is defined as a method to harness natural selection in order to engineer proteins to acquire particular properties that are not associated with the protein in nature. Literature has provided numerous examples regarding the implementation of directed evolution to successfully alter molecular specificity and catalysis¹. The primary advantage of utilizing directed evolution instead of more rational-based approaches for molecular engineering relates to the volume and diversity of variants that can be screened². One possible application of directed evolution involves improving structural stability of bacteriolytic enzymes, such as endolysins. Bacteriophage encode and express endolysins to hydrolyze a critical covalent bond in the peptidoglycan (i.e. cell wall) of bacteria, resulting in host cell lysis and liberation of progeny virions. Notably, these enzymes possess the ability to extrinsically induce lysis to susceptible bacteria in the absence of phage and furthermore have been validated both in vitro and in vivo for their therapeutic potential^3-5. The subject of our directed evolution study involves the PlyC endolysin, which is composed of PlyCA and PlyCB subunits⁶. When purified and added extrinsically, the PlyC holoenzyme lyses group A streptococci (GAS) as well as other streptococcal groups in a matter of seconds and furthermore has been validated in vivo against GAS⁷. Significantly, monitoring residual enzyme kinetics after elevated temperature incubation provides distinct evidence that PlyC loses lytic activity abruptly at 45 °C, suggesting a short therapeutic shelf life, which may limit additional development of this enzyme. Further studies reveal the lack of thermal stability is only observed for the PlyCA subunit, whereas the PlyCB subunit is stable up to ~90 °C (unpublished observation). In addition to PlyC, there are several examples in literature that describe the thermolabile nature of endolysins. For example, the Staphylococcus aureus endolysin LysK and Streptococcus pneumoniae endolysins Cpl-1 and Pal lose activity spontaneously at 42 °C, 43.5 °C and 50.2 °C, respectively^8-10. According to the Arrhenius equation, which relates the rate of a chemical reaction to the temperature present in the particular system, an increase in thermostability will correlate with an increase in shelf life expectancy¹¹. Toward this end, directed evolution has been shown to be a useful tool for altering the thermal activity of various molecules in nature, but never has this particular technology been exploited successfully for the study of bacteriolytic enzymes. Likewise, successful accounts of progressing the structural stability of this particular class of antimicrobials altogether are nonexistent. In this video, we employ a novel methodology that uses an error-prone DNA polymerase followed by an optimized screening process using a 96 well microtiter plate format to identify mutations to the PlyCA subunit of the PlyC streptococcal endolysin that correlate to an increase in enzyme kinetic stability (Figure 1). Results after just one round of random mutagenesis suggest the methodology is generating PlyC variants that retain more than twice the residual activity when compared to wild-type (WT) PlyC after elevated temperature treatment.     

Density dependent selection 1: A stable feasible equilibrium may not be attainable

The joint evolution of gene frequency p, and population size N is studied. It is shown that when the genotypic fitnesses are logistic functions of the population size, sets of initial states exist which lead to bizarre behavior. Even though equilibria may be locally stable, these sets of initial conditions eventually produce negative fitnesses. Alternative models are discussed as are some general properties of the mean fitness.     

Maximizing the precision of the mathematically ideal radioimmunoassay at a specified dose

Mathematical analysis of the equation for the radioimmunoassay, R²+R(1+Kp+Kp^*–Kq)–Kq=0, gives a general solution for the dose of maximum precision, ²R/p², of the assay. This solution, (R–c₁+Kp)(R–c₂), and the radioimmunoassay equation can be used to determine theoretical conditions for maximum precision of a radioimmunoassay at any dose p. A theoretically optimal antibody concentration, q, can be calculated for precision at a critical concentration of interest, p. This is dependent on the affinity constant, K, of the particular antibody being used and, potentially, the amount of labeled tracer, p^*. This solution idealizes the radioimmunoassay to a one affinity constant system. Though this may be achievable, particularly with monoclonal antibodies or receptors, the applicability needs to be explored experimentally.     

Travelling wave solutions of diffusive Lotka-Volterra equations

We establish the existence of travelling wave solutions for two reaction diffusion systems based on the Lotka-Volterra model for predator and prey interactions. For simplicity, we consider only 1 space dimension. The waves are of transition front type, analogous to the travelling wave solutions discussed by Fisher and Kolmogorov et al. for a scalar reaction diffusion equation. The waves discussed here are not necessarily monotone. For any speed c there is a travelling wave solution of transition front type. For one of the systems discussed here, there is a distinguished speed c^* dividing the waves into two types, waves of speed c < c^* being one type, waves of speed c ? c^* being of the other type. We present numerical evidence that for this system the wave of speed c^* is stable, and that c^* is an asymptotic speed of propagation in some sense. For the other system, waves of all speeds are in some sense stable. The proof of existence uses a shooting argument and a Lyapunov function. We also discuss some possible biological implications of the existence of these waves.