Signatures of neutral evolution in exponentially growing tumors: A theoretical perspective

Recent work of Sottoriva, Graham, and collaborators have led to the controversial claim that exponentially growing tumors have a site frequency spectrum that follows the 1/f law consistent with neutral evolution. This conclusion has been criticized based on data quality issues, statistical considerations, and simulation results. Here, we use rigorous mathematical arguments to investigate the site frequency spectrum in the two-type model of clonal evolution. If the fitnesses of the two types are λ₀ < λ₁, then the site frequency spectrum is c/f^α where α = λ₀/λ₁. This is due to the advantageous mutations that produce the founders of the type 1 population. Mutations within the growing type 0 and type 1 populations follow the 1/f law. Our results show that, in contrast to published criticisms, neutral evolution in an exponentially growing tumor can be distinguished from the two-type model using the site frequency spectrum.     

Random walk in genome space: A key ingredient of intermittent dynamics of community assembly on evolutionary time scales

Community assembly is studied using individual-based multispecies models. The models have stochastic population dynamics with mutation, migration, and extinction of species. Mutants appear as a result of mutation of the resident species, while migrants have no correlation with the resident species. It is found that the dynamics of community assembly with mutations are quite different from the case with migrations. In contrast to mutation models, which show intermittent dynamics of quasi-steady states interrupted by sudden reorganizations of the community, migration models show smooth and gradual renewal of the community. As a consequence, instead of the 1/f diversity fluctuations found for the mutation models, 1/f², random-walk like fluctuations are observed for the migration models. In addition, a characteristic species-lifetime distribution is found: a power law that is cut off by a “skewed” distribution in the long-lifetime regime. The latter has a longer tail than a simple exponential function, which indicates an age-dependent species-mortality function. Since this characteristic profile has been observed, both in fossil data and in several other mathematical models, we conclude that it is a universal feature of macroevolution.     

On the mathematical basis of the variance-mean power relationship

The mathematical basis of a widely-known variance-mean power relationship of ecological populations was examined. It is shown that the log variance (S ²)—log mean, (m) plot is virtually delimited by two lines logS ²=logn+2 logm and logS ²=logm, thus increasing the chance that a linear regression line can be successfully fitted, without a profoundly behavioural background. This makes difficult the task of interpreting a successful fit of the power law regression and its parameterb in a biologically meaningful manner. In comparison with the power law regression, Iwao'sm ^*-m regression is structurally less constrained, i.e. has a wider spatial region in which data points can scatter. This suggests that a comparison between the two methods in terms of how good a fit is achieved for a particular data set is largely meaningless, since the power law regression may inherently produce a better fit due to its constrained spatial entity. Furthermore, it could be argued that a successful fit in Iwao's method, when found, is less taxed with mathematical arterfacts and perhaps more clearly linked to some biological mechanisms underlying spatial dispersion of populations.     

Alpha blocking and 1/fβ spectral scaling in resting EEG can be accounted for by a sum of damped alpha band oscillatory processes

The dynamical and physiological basis of alpha band activity and 1/f^β noise in the EEG are the subject of continued speculation. Here we conjecture, on the basis of empirical data analysis, that both of these features may be economically accounted for through a single process if the resting EEG is conceived of being the sum of multiple stochastically perturbed alpha band damped linear oscillators with a distribution of dampings (relaxation rates). The modulation of alpha-band and 1/f^β noise activity by changes in damping is explored in eyes closed (EC) and eyes open (EO) resting state EEG. We aim to estimate the distribution of dampings by solving an inverse problem applied to EEG power spectra. The characteristics of the damping distribution are examined across subjects, sensors and recording condition (EC/EO). We find that there are robust changes in the damping distribution between EC and EO recording conditions across participants. The estimated damping distributions are found to be predominantly bimodal, with the number and position of the modes related to the sharpness of the alpha resonance and the scaling (β) of the power spectrum (1/f^β). The results suggest that there exists an intimate relationship between resting state alpha activity and 1/f^β noise with changes in both governed by changes to the damping of the underlying alpha oscillatory processes. In particular, alpha-blocking is observed to be the result of the most weakly damped distribution mode becoming more heavily damped. The results suggest a novel way of characterizing resting EEG power spectra and provides new insight into the central role that damped alpha-band activity may play in characterising the spatio-temporal features of resting state EEG.     

Quantification of emotion by nonlinear analysis of the chaotic dynamics of electroencephalograms during perception of 1/f music

The goal of this study is to quantify and determine the way in which the emotional response to music is reflected in the electrical activities of the brain. When the power spectrum of sequences of musical notes is inversely proportional to the frequency on a log-log plot, we call it 1/f music. According to previous research, most listeners agree that 1/f music is much more pleasing than white (1/f ⁰) or brown (1/f ²) music. Based on these studies, we used nonlinear methods to investigate the chaotic dynamics of electroencephalograms (EEGs) elicited by computer-generated 1/f music, white music, and brown music. In this analysis, we used the correlation dimension and the largest Lyapunov exponent as measures of complexity and chaos. We developed a new method that is strikingly faster and more accurate than other algorithms for calculating the nonlinear invariant measures from limited noisy data. At the right temporal lobe, 1/f music elicited lower values of both the correlation dimension and the largest Lyapunov exponent than white or brown music. We observed that brains which feel more pleased show decreased chaotic electrophysiological behavior. By observing that the nonlinear invariant measures for the 1/f distribution of the rhythm with the melody kept constant are lower than those for the 1/f distribution of melody with the rhythm kept constant, we could conclude that the rhythm variations contribute much more to a pleasing response to music than the melody variations do. These results support the assumption that chaos plays an important role in brain function, especially emotion. Received: 30 December 1996 / Accepted in revised form: 18 December 1997     

General relation between variance-time curve and power spectral density for point processes exhibiting 1/f β-fluctuations,with special reference to heart rate variability

Counting statistics in the form of the variance-time curve provides an alternative to spectral analysis for point processes exhibiting 1/f ^β-fluctuations, such as the heart beat. However, this is true only for β<1. Here, the case of general β is considered. To that end, the mathematical relation between the variance-time curve and power spectral density in the presence of 1/f ^β-noise is worked out in detail. A modified version of the variance-time curve is presented, which allows us to deal also with the case β?1. Some applications to the analysis of heart rate variability are given.     

Neuronal Shot Noise and Brownian 1/f2 Behavior in the Local Field Potential

We demonstrate that human electrophysiological recordings of the local field potential (LFP) from intracranial electrodes, acquired from a variety of cerebral regions, show a ubiquitous 1/f² scaling within the power spectrum. We develop a quantitative model that treats the generation of these fields in an analogous way to that of electronic shot noise, and use this model to specifically address the cause of this 1/f² Brownian noise. The model gives way to two analytically tractable solutions, both displaying Brownian noise: 1) uncorrelated cells that display sharp initial activity, whose extracellular fields slowly decay in time and 2) rapidly firing, temporally correlated cells that generate UP-DOWN states.     

The theoretical possibility of reverse translation of proteins into genes

It is shown on a theoretical basis that the existence of a “power law” relationship between body mass M and total metabolic heat generation rate Q of the form Q = kM^α does not uniquely determine the dependence of metabolic rate on body temperature. However, it is shown that a particular assumption for this temperature dependence, successful in other problems, does predict a “power law” similar to the empirical one. At the same time it also accounts satisfactorily for the linear dependence of metabolic rate on ambient temperature.     

Mammalian Fbh1 is important to restore normal mitotic progression following decatenation stress

We have addressed the role of the F-box helicase 1 (Fbh1) protein during genome maintenance in mammalian cells. For this, we generated two mouse embryonic stem cell lines deficient for Fbh1: one with a homozygous deletion of the N-terminal F-box domain (Fbh1^f/f), and the other with a homozygous disruption (Fbh1^?/?). Consistent with previous reports of Fbh1-deficiency in vertebrate cells, we found that Fbh1^?/? cells show a moderate increase in Rad51 localization to DNA damage, but no clear defect in chromosome break repair. In contrast, we found that Fbh1^f/f cells show a decrease in Rad51 localization to DNA damage and increased cytoplasmic localization of Rad51. However, these Fbh1^f/f cells show no clear defects in chromosome break repair. Since some Rad51 partners and F-box-associated proteins (Skp1-Cul1) have been implicated in progression through mitosis, we considered whether Fbh1 might play a role in this process. To test this hypothesis, we disrupted mitosis using catalytic topoisomerase II inhibitors (bisdioxopiperazines), which inhibit chromosome decatenation. We found that both Fbh1^f/f and Fbh1^?/? cells show hypersensitivity to topoisomerase II catalytic inhibitors, even though the degree of decatenation stress was not affected. Furthermore, following topoisomerase II catalytic inhibition, both Fbh1-deficient cell lines show substantial defects in anaphase separation of chromosomes. These results indicate that Fbh1 is important for restoration of normal mitotic progression following decatenation stress.     

Evidence that chlorophyll f functions solely as an antenna pigment in far-red-light photosystem I from Fischerella thermalis PCC 7521

The Photosystem I (PSI) reaction center in cyanobacteria is comprised of ~96 chlorophyll (Chl) molecules, including six specialized Chl molecules denoted Chl1A/Chl1B (P₇₀₀), Chl2A/Chl2B, and Chl3A/Chl3B that are arranged in two branches and function in primary charge separation. It has recently been proposed that PSI from Chroococcidiopsis thermalis (Nürnberg et al. (2018) Science 360, 1210–1213) and Fischerella thermalis PCC 7521 (Hastings et al. (2019) Biochim. Biophys. Acta 1860, 452–460) contain Chl f in the positions Chl2A/Chl2B. We tested this proposal by exciting RCs from white-light grown (WL-PSI) and far-red light grown (FRL-PSI) F. thermalis PCC 7521 with femtosecond pulses and analyzing the optical dynamics. If Chl f were in the position Chl2A/Chl2B in FRL-PSI, excitation at 740 nm should have produced the charge-separated state P₇₀₀⁺A₀⁻ followed by electron transfer to A₁ with a τ of ≤25 ps. Instead, it takes ~230 ps for the charge-separated state to develop because the excitation migrates uphill from Chl f in the antenna to the trapping center. Further, we observe a strong electrochromic shift at 685 nm in the final P₇₀₀⁺A₁⁻ spectrum that can only be explained if Chl a is in the positions Chl2A/Chl2B. Similar arguments rule out the presence of Chl f in the positions Chl3A/Chl3B; hence, Chl f is likely to function solely as an antenna pigment in FRL-PSI. We additionally report the presence of an excitonically coupled homo- or heterodimer of Chl f absorbing around 790 nm that is kinetically independent of the Chl f population that absorbs around 740 nm.     

Multifractal analysis of K+ channel activity

The Multifractal version of the Detrended Fluctuation Analysis was used for the study of non-stationary dwell time series of Ca²⁺-activated K⁺ channels (K_Ca channels) in cultured kidney Vero cells and of voltage-dependent K⁺ channels (K_v channels) in mollusc (Lymnaea stagnalis) neurons. The data obtained can briefly be summarized as follows: (i) The generalized fluctuation function F _q (l) strongly depends on the index (order) q; for monofractal time series, such dependence is nonexistent; (ii) The relationship between the scaling exponent τ commonly employed in standard multifractal analysis and q is characterized by two slopes and a transitory region, whereas monofractal processes are characterized by the linear dependence; (iii) The relationship between the singularity spectrum f(h) and the Hurst exponent h is bell-shaped, while in the case of monofractal processes it is represented by a single point f(h) = 1. Random mixing of the time series resulted in the narrowing of the spectrum f(h) and a shift of f(h) towards the value more characteristic of stochastic (monofractal) processes (h ～ 0.5). It is concluded that the activities of both K_Ca channels in kidney Vero cells and of K_V channels in mollusc (Lymnaea stagnalis) neurons can be characterized as multifractal processes.     

1/f noise in black lipid membranes induced by ionic channels formed by chemically dimerized gramicidin A

Summary The noise behavior of lipid bilayer membranes, doped with a chemically dimerized gramicidin A, was investigated. In contrast to normal gramicidin A, which generates a Lorentzian type power spectrum due to the formation and disappearance of conducting dimers, the current power spectrum densityS _m (f) obtained with this gramicidin A derivative showed over several orders of magnitude a clear 1/f behavior. The intensity of this 1/f component was analyzed as a function of the membrane-applied voltage, membrane resistance, electrolyte concentration, and composition. The relationship between the meansquare fluctuation in current and the membrane current mean value was found to follow Hooge's equation, i.e., I ²=I _m ² /N f whereN is the number of channels and is a constant equal to 1.0×10^–2. It is suggested that a 1/f type noise was observed because the chemically dimerized form of gramicidin A produces long lasting cation selective channels.     

Macroscopic Models of Local Field Potentials and the Apparent 1/f Noise in Brain Activity

The power spectrum of local field potentials (LFPs) has been reported to scale as the inverse of the frequency, but the origin of this 1/f noise is at present unclear. Macroscopic measurements in cortical tissue demonstrated that electric conductivity (as well as permittivity) is frequency-dependent, while other measurements failed to evidence any dependence on frequency. In this article, we propose a model of the genesis of LFPs that accounts for the above data and contradictions. Starting from first principles (Maxwell equations), we introduce a macroscopic formalism in which macroscopic measurements are naturally incorporated, and also examine different physical causes for the frequency dependence. We suggest that ionic diffusion primes over electric field effects, and is responsible for the frequency dependence. This explains the contradictory observations, and also reproduces the 1/f power spectral structure of LFPs, as well as more complex frequency scaling. Finally, we suggest a measurement method to reveal the frequency dependence of current propagation in biological tissue, and which could be used to directly test the predictions of this formalism.     

Bounds on the total population for species governed by reaction-diffusion equations in arbitrary two-dimensional regions

Analysis based on the integration of differential inequalities is employed to derive upper and lower bounds on the total populationN(t) = ∫ _R θ(x ₁,x ₂,t) dx ₁ dx ₂ of a biological species with an area-density distribution function θ=θ(x ₁,x ₂,t) (≥0) governed by a reaction-diffusion equation of the form ∂θ/∂t =D∇²θ +fθ −gθ ⁿ⁺¹ whereD (>0),n (>0),f andg are constant parameters, θ=0 at all points on the boundary ∂R of an (arbitrary) two-dimensional regionR, and the initial distribution (θ(_{x 1},x ₂, 0) is such thatN(0) is finite. Forg≥0 withR the entire two-dimensional Euclidean space, a lower bound onN(t) is obtained, showing in particular thatN(∞) is bounded below by a finite positive quantity forf≥0 andn>1. An upper bound onN(t) is obtained for arbitrary bounded or unbounded)R withn=1,f andg negative, and ∫ _R θ(x ₁,x ₂, 0)² dx ₁ dx ₂ sufficiently small in magnitude, implying that the population goes to extinction with increasing values of the time,N(∞)=0. Forg≥0 andR of finite area, the analysis yields upper bounds onN(t), predicting eventual extinction of the population if eitherf≤0 or if the area ofR is less than a certain grouping of the parameters in cases for whichf is positive. These results are directly applicable to biological species with distributions satisfying the Fisher equation in two spatial dimensions and to species governed by certain specialized population models.     

Recording Single Neurons' Action Potentials from Freely Moving Pigeons Across Three Stages of Learning

While the subject of learning has attracted immense interest from both behavioral and neural scientists, only relatively few investigators have observed single-neuron activity while animals are acquiring an operantly conditioned response, or when that response is extinguished. But even in these cases, observation periods usually encompass only a single stage of learning, i.e. acquisition or extinction, but not both (exceptions include protocols employing reversal learning; see Bingman et al.¹ for an example). However, acquisition and extinction entail different learning mechanisms and are therefore expected to be accompanied by different types and/or loci of neural plasticity.Accordingly, we developed a behavioral paradigm which institutes three stages of learning in a single behavioral session and which is well suited for the simultaneous recording of single neurons'' action potentials. Animals are trained on a single-interval forced choice task which requires mapping each of two possible choice responses to the presentation of different novel visual stimuli (acquisition). After having reached a predefined performance criterion, one of the two choice responses is no longer reinforced (extinction). Following a certain decrement in performance level, correct responses are reinforced again (reacquisition). By using a new set of stimuli in every session, animals can undergo the acquisition-extinction-reacquisition process repeatedly. Because all three stages of learning occur in a single behavioral session, the paradigm is ideal for the simultaneous observation of the spiking output of multiple single neurons. We use pigeons as model systems, but the task can easily be adapted to any other species capable of conditioned discrimination learning.     

Spectroscopic studies of the iron and manganese reconstituted tyrosyl radical in Bacillus cereus ribonucleotide reductase R2 protein

Ribonucleotide reductase (RNR) catalyzes the rate limiting step in DNA synthesis where ribonucleotides are reduced to the corresponding deoxyribonucleotides. Class Ib RNRs consist of two homodimeric subunits: R1E, which houses the active site; and R2F, which contains a metallo cofactor and a tyrosyl radical that initiates the ribonucleotide reduction reaction. We studied the R2F subunit of B. cereus reconstituted with iron or alternatively with manganese ions, then subsequently reacted with molecular oxygen to generate two tyrosyl-radicals. The two similar X-band EPR spectra did not change significantly over 4 to 50 K. From the 285 GHz EPR spectrum of the iron form, a g ₁-value of 2.0090 for the tyrosyl radical was extracted. This g ₁-value is similar to that observed in class Ia E. coli R2 and class Ib R2Fs with iron-oxygen cluster, suggesting the absence of hydrogen bond to the phenoxyl group. This was confirmed by resonance Raman spectroscopy, where the stretching vibration associated to the radical (C-O, ν_7a = 1500 cm⁻¹) was found to be insensitive to deuterium-oxide exchange. Additionally, the ¹⁸O-sensitive Fe-O-Fe symmetric stretching (483 cm⁻¹) of the metallo-cofactor was also insensitive to deuterium-oxide exchange indicating no hydrogen bonding to the di-iron-oxygen cluster, and thus, different from mouse R2 with a hydrogen bonded cluster. The HF-EPR spectrum of the manganese reconstituted RNR R2F gave a g ₁-value of ∼2.0094. The tyrosyl radical microwave power saturation behavior of the iron-oxygen cluster form was as observed in class Ia R2, with diamagnetic di-ferric cluster ground state, while the properties of the manganese reconstituted form indicated a magnetic ground state of the manganese-cluster. The recent activity measurements (Crona et al., (2011) J Biol Chem 286: 33053–33060) indicates that both the manganese and iron reconstituted RNR R2F could be functional. The manganese form might be very important, as it has 8 times higher activity.     

Fractal analysis of surface electromyography signals: A novel power spectrum-based method

This paper presents a novel power spectrum-based method for fractal analysis of surface electromyography signals. This method, named the bi-phase power spectrum method, provides a bi-phase power-law which represents a multi-scale statistically self-affine signal. This form of statistical self-affinity provides an accurate approximation for stochastic signals originating from a strong non-linear combination of a number of similar distributions, such as surface electromyography signals which are formed by the summation of a number of single muscle fiber action potentials. This power-law is characterized by a set of spectral indicators, which are related to distributional and geometrical characteristics of the electromyography signal’s interference pattern. These novel spectral indicators are capable of sensing the effects of motor units’ recruitment and shape separately by exploiting the geometry of the interference pattern. The bi-phase power spectrum method is compared to geometrical techniques and the 1/f^α approach for fractal analysis of electromyography signals. The extracted indicators using the bi-phase power spectrum method are evaluated in the context of force and joint angle and the results of a human study are presented. Results demonstrate that the bi-phase power spectrum method provides reliable information, consisting of components capable of sensing force and joint angle effects separately, which could be used as complementary information for confounded conventional measures.     

Investigations on the causes of low susceptibility of scentless mayweed to 2,4-D

White mustard (Sinapis alba L.) plants, susceptible to 2,4-D, and scentless mayweed (Tripleurospermum maritimum L.) plants, resistant to 2,4-D, markedly differ in the distribution and metabolism of 2,4-D. When 2,4-D-¹⁴C was applied onto the leaf, radioactivity was found in mustard after 8 days in the entire plant, whereas in scentless mayweed radioactivity occurred mainly in the leaf onto which it was applied and in another one or two leaves, with the roots showing only traces of radioactivity. The two plant species differed both in the character and in the number and amount of metabolites detected in aqueous and ether fractions. The metabolite with R_f 0.63 -0.65, soluble in ethylether (39% of the total radioactivity applied), and that with R_f 0.35 -0.36 (23% of the total radioactivity) were present exclusively in the roots of scentless mayweed plants. The R_f values established indicate that the former metabolite may be a conjugate with the amino acids alanine and valine and the latter a hydroxylated derivate of 2,4-D. In the shoots of both plants, 2,4-D-¹⁴C was metabolized to identical metabolites with R_f 0.59 -0.61, which occurred in mustard plants only in trace amounts. Free 2,4-D-¹⁴C occurred in the shoots of both plants and in considerable amounts also in mustard roots; it could not be demonstrated in the roots of scentless mayweed. The two species did not differ in the uptake rate or in the amount of absorbed 2,4-D-¹⁴C.     

Competitive blocking of apical sodium channels in epithelia

Apical sodium-selective channels in frog skin, when blocked by amiloride or triamterene, exhibit fluctuations in current, the spectra of which are Lorentzian. These effects have been modeled previously with two-state and three-state models by Lindemann and Van Driessche. A recent observation by Hoshiko and Van Driessche that corner frequencies are lowered by increasing the apical sodium concentration cannot be accounted for by these models. We explore the possibility that sodium (S) and amiloride (A) compete for a site at the mouth of the channel. A new three-state channel model (sodium-occupied, open/unoccupied, open/amiloride-blocked) is analyzed. Its corner frequency is of the form f_c = f_co[1 + (A/K_A)/(1 + S/K_S)], consistent with the observed sodium dependence of the corner frequency. The minimum frequency, f_co, and the inhibition constants, K_A and K_S, are expressed in terms of the rate constants of the model. To account for sodium self-inhibition, we postulate that two sodium ions in the channel may result in clogging — a fourth state. The two corner frequencies are calculated; so are the plateau values of the noise power. The noise power shows a maximum as a function of blocker concentration, as observed previously using triamterene. The four-state model predicts the observed suppression by small amounts of blocker of the low-frequency sodium (clogging) noise.