Spontaneous regression of residual tumour burden: prediction by Monte Carlo simulation.

Current cancer treatment protocols are designed to release the tumour burden down to a small number of cells. In this study, we use Monte Carlo simulations to show that small populations of cells with intrinsic cell loss rates comparable to the cell loss rates observed clinically in human tumours, may regress spontaneously. Large populations of cells tend to grow under the same conditions of cell loss that result in extinction of small clones. Furthermore, minor variations in the intrinsic cell death probability near 0.50 result in large differences in the number of surviving cells calculated at the 100th generation. When Monte Carlo simulations of clonal growth resulted in clones with large populations (> 50 cells), the population as a whole behaved in a deterministic fashion (logarithmic growth) similar to those observed in clinically observed neoplasms and consistent with other published models of tumour growth. These findings provide a plausible explanation for the clinically observed failure of tumours to recur in instances where tumour burden remains following cancer therapy. The findings also demonstrate the usefulness of the Monte Carlo method to simulate biologic events in populations where the fate of each member of a population can be modeled probabilistically.     

Survival of small populations under demographic stochasticity.

We estimate the mean time to extinction of small populations in an environment with constant carrying capacity but under stochastic demography. In particular, we investigate the interaction of stochastic variation in fecundity and sex ratio under several different schemes of density dependent population growth regimes. The methods used include Markov chain theory, Monte Carlo simulations, and numerical simulations based on Markov chain theory. We find a strongly enhanced extinction risk if stochasticity in sex ratio and fluctuating population size act simultaneously as compared to the case where each mechanism acts alone. The distribution of extinction times deviates slightly from a geometric one, in particular for short extinction times. We also find that whether maximization of intrinsic growth rate decreases the risk of extinction or not depends strongly on the population regulation mechanism. If the population growth regime reduces populations above the carrying capacity to a size below the carrying capacity for large r (overshooting) then the extinction risk increases if the growth rate deviates from an optimal r-value.     

Promotion of initiated cells by radiation-induced cell inactivation

Cells on the way to carcinogenesis can have a growth advantage relative to normal cells. It has been hypothesized that a radiation-induced growth advantage of these initiated cells might be induced by an increased cell replacement probability of initiated cells after inactivation of neighboring cells by radiation. Here Monte Carlo simulations extend this hypothesis for larger clones: The effective clonal expansion rate decreases with clone size. This effect is stronger for the two-dimensional than for the three-dimensional situation. The clones are irregular, far from a circular shape. An exposure-rate dependence of the effective clonal expansion rate could come in part from a minimal recovery time of the initiated cells for symmetric cell division.     

Extinction risk of a density-dependent population estimated from a time series of population size

Environmental threats, such as habitat size reduction or environmental pollution, may not cause immediate extinction of a population but shorten the expected time to extinction. We develop a method to estimate the mean time to extinction for a density-dependent population with environmental fluctuation. We first derive a formula for a stochastic differential equation model (canonical model) of a population with logistic growth with environmental and demographic stochasticities. We then study an approximate maximum likelihood (AML) estimate of three parameters (intrinsic growth rate r, carrying capacity K, and environmental stochasticity sigma(2)(e)) from a time series of population size. The AML estimate of r has a significant bias, but by adopting the Monte Carlo method, we can remove the bias very effectively (bias-corrected estimate). We can also determine the confidence interval of the parameter based on the Monte Carlo method. If the length of the time series is moderately long (with 40-50 data points), parameter estimation with the Monte Carlo sampling bias correction has a relatively small variance. However, if the time series is short (less than or equal to 10 data points), the estimate has a large variance and is not reliable. If we know the intrinsic growth rate r, however, the estimate of K and sigma(2)(e)and the mean extinction time T are reliable even if only a short time series is available. We illustrate the method using data for a freshwater fish, Japanese crucian carp (Carassius auratus subsp.) in Lake Biwa, in which the growth rate and environmental noise of crucian carp are estimated using fishery records.     

Folding simulations of small proteins

Understanding how a protein folds is a long-standing challenge in modern science. We have used an optimized atomistic model (united-residue force field) to simulate folding of small proteins of various structures: HP-36 (alpha protein), protein A (beta), 1fsd (alpha+beta), and betanova (beta). Extensive Monte Carlo folding simulations (ten independent runs with 10(9) Monte Carlo steps at a temperature) starting from non-native conformations are carried out for each protein. In all cases, proteins fold into their native-like conformations at appropriate temperatures, and glassy transitions occur at low temperatures. To investigate early folding trajectories, 200 independent runs with 10(6) Monte Carlo steps are also performed at a fixed temperature for a protein. There are a variety of possible pathways during non-equilibrium early processes (fast process, approximately 10(4) Monte Carlo steps). Finally, these pathways converge to the point unique for each protein. The convergence point of the early folding pathways can be determined only by direct folding simulations. The free energy surface, an equilibrium thermodynamic property, dictates the rest of the folding (slow process, approximately 10(8) Monte Carlo steps).     

Parallel tempering simulations of HP-36

We report results from all-atom Monte Carlo simulations of the 36-residue villin headpiece subdomain HP-36. Protein-solvent interactions are approximated by an implicit solvent model. The parallel tempering is used to overcome the problem of slow convergence in low-temperature protein simulations. Our results show that this technique allows one to sample native-like structures of small proteins and points out the need for improved energy functions.     

Model Prediction and Validation of an Order Mechanism Controlling the Spatiotemporal Phenotype of Early Hepatocellular Carcinoma

Recently, hepatocyte–sinusoid alignment (HSA) has been identified as a mechanism that supports the coordination of hepatocytes during liver regeneration to reestablish a functional micro-architecture (Hoehme et al. in Proc Natl Acad Sci 107(23):10371–10376, 2010). HSA means that hepatocytes preferentially align along the closest micro-vessels. Here, we studied whether this mechanism is still active in early hepatocellular tumors. The same agent-based spatiotemporal model that previously correctly predicted HSA in liver regeneration was further developed to simulate scenarios in early tumor development, when individual initiated hepatocytes gain increased proliferation capacity. The model simulations were performed under conditions of realistic liver micro-architectures obtained from 3D reconstructions of confocal laser scanning micrographs. Interestingly, the established model predicted that initiated hepatocytes at first arrange in elongated patterns. Only when the tumor progresses to cell numbers of approximately 4000, does it adopt spherical structures. This prediction may have relevant consequences, since elongated tumors may reach critical structures faster, such as larger vessels, compared to a spherical tumor of similar cell number. Interestingly, this model prediction was confirmed by analysis of the spatial organization of initiated hepatocytes in a rat liver tumor initiation study using single doses of 250 mg/kg of the genotoxic carcinogen N-nitrosomorpholine (NNM). Indeed, small clusters of GST-P positive cells induced by NNM were elongated, almost columnar, while larger GDT-P positive foci of approximately the size of liver lobuli adopted spherical shapes. From simulations testing numerous possible mechanisms, only HSA could explain the experimentally observed initial deviation from spherical shape. The present study demonstrates that the architecture of small cell clusters of hepatocytes early after initiation is still controlled by physiological mechanisms. However, this coordinating influence is lost when the tumor grows to approximately 4000 cells, leading to further growth in spherical shape. Our findings stress the potential importance of organ micro-architecture in understanding tumor phenotypes.     

Information dynamics in carcinogenesis and tumor growth

The storage and transmission of information is vital to the function of normal and transformed cells. We use methods from information theory and Monte Carlo theory to analyze the role of information in carcinogenesis. Our analysis demonstrates that, during somatic evolution of the malignant phenotype, the accumulation of genomic mutations degrades intracellular information. However, the degradation is constrained by the Darwinian somatic ecology in which mutant clones proliferate only when the mutation confers a selective growth advantage. In that environment, genes that normally decrease cellular proliferation, such as tumor suppressor or differentiation genes, suffer maximum information degradation. Conversely, those that increase proliferation, such as oncogenes, are conserved or exhibit only gain of function mutations. These constraints shield most cellular populations from catastrophic mutator-induced loss of the transmembrane entropy gradient and, therefore, cell death. The dynamics of constrained information degradation during carcinogenesis cause the tumor genome to asymptotically approach a minimum information state that is manifested clinically as dedifferentiation and unconstrained proliferation. Extreme physical information (EPI) theory demonstrates that altered information flow from cancer cells to their environment will manifest in-vivo as power law tumor growth with an exponent of size 1.62. This prediction is based only on the assumption that tumor cells are at an absolute information minimum and are capable of "free field" growth that is, they are unconstrained by external biological parameters. The prediction agrees remarkably well with several studies demonstrating power law growth in small human breast cancers with an exponent of 1.72+/-0.24. This successful derivation of an analytic expression for cancer growth from EPI alone supports the conceptual model that carcinogenesis is a process of constrained information degradation and that malignant cells are minimum information systems. EPI theory also predicts that the estimated age of a clinically observed tumor is subject to a root-mean square error of about 30%. This is due to information loss and tissue disorganization and probably manifests as a randomly variable lag phase in the growth pattern that has been observed experimentally. This difference between tumor size and age may impose a fundamental limit on the efficacy of screening based on early detection of small tumors. Independent of the EPI analysis, Monte Carlo methods are applied to predict statistical tumor growth due to perturbed information flow from the environment into transformed cells. A "simplest" Monte Carlo model is suggested by the findings in the EPI approach that tumor growth arises out of a minimally complex mechanism. The outputs of large numbers of simulations show that (a) about 40% of the populations do not survive the first two-generations due to mutations in critical gene segments; but (b) those that do survive will experience power law growth identical to the predicted rate obtained from the independent EPI approach. The agreement between these two very different approaches to the problem strongly supports the idea that tumor cells regress to a state of minimum information during carcinogenesis, and that information dynamics are integrally related to tumor development and growth.     

A multiscale model for avascular tumor growth

The desire to understand tumor complexity has given rise to mathematical models to describe the tumor microenvironment. We present a new mathematical model for avascular tumor growth and development that spans three distinct scales. At the cellular level, a lattice Monte Carlo model describes cellular dynamics (proliferation, adhesion, and viability). At the subcellular level, a Boolean network regulates the expression of proteins that control the cell cycle. At the extracellular level, reaction-diffusion equations describe the chemical dynamics (nutrient, waste, growth promoter, and inhibitor concentrations). Data from experiments with multicellular spheroids were used to determine the parameters of the simulations. Starting with a single tumor cell, this model produces an avascular tumor that quantitatively mimics experimental measurements in multicellular spheroids. Based on the simulations, we predict: 1), the microenvironmental conditions required for tumor cell survival; and 2), growth promoters and inhibitors have diffusion coefficients in the range between 10(-6) and 10(-7) cm2/h, corresponding to molecules of size 80-90 kDa. Using the same parameters, the model also accurately predicts spheroid growth curves under different external nutrient supply conditions.     

Monte Carlo estimates of natural variation in HIV infection

We describe a Monte Carlo simulation of the within-host dynamics of human immunodeficiency virus 1 (HIV-1). The simulation proceeds at the level of individual T-cells and virions in a small volume of plasma, thus capturing the inherent stochasticity in viral replication, mutation and T-cell infection. When cell lifetimes are distributed exponentially in the Monte Carlo approach, our simulation results are in perfect agreement with the predictions of the corresponding systems of differential equations from the literature. The Monte Carlo model, however, uniquely allows us to estimate the natural variability in important parameters such as the T-cell count, viral load, and the basic reproductive ratio, in both the presence and absence of drug therapy. The simulation also yields the probability that an infection will not become established after exposure to a viral inoculum of a given size. Finally, we extend the Monte Carlo approach to include distributions of cell lifetimes that are less-dispersed than exponential.     

Bayesian modeling of embryonic growth using latent variables

In a growth model, individuals move progressively through a series of states in which each state is indicative of developmental status. Interest lies in estimating the rate of progression through each state while incorporating covariates that might affect the transition rates. We develop a Bayesian discrete-time multistate growth model for inference from cross-sectional data with unknown initiation times. For each subject, data are collected at only one time point at which we observe the state as well as covariates that measure developmental progress. We link the developmental progress variables to an underlying latent growth variable that can also affect the state transition rates. A subject with slow latent growth will then have relatively small developmental progress covariates and move through state transitions slowly. We then examine the association between latent growth and the probability of future events in a novel study of embryonic development and pregnancy loss. Using a Markov chain Monte Carlo (MCMC) algorithm for posterior computation, we found evidence in favor of a previously hypothesized but unproven association between slow growth early in pregnancy and increased risk of future spontaneous abortion.     

Allee effects and extinction in a lattice model

In the interest of conservation, the importance of having a large habitat available for a species is widely known. Here, we introduce a lattice-based model for a population and look at the importance of fluctuations as well as that of the population density, particularly with respect to Allee effects. We examine the model analytically and by Monte Carlo simulations and find that, while the size of the habitat is important, there exists a critical population density below which the probability of extinction is greatly increased. This has large consequences with respect to conservation, especially in the design of habitats and for populations whose density has become small. In particular, we find that the probability of survival for small populations can be increased by a reduction in the size of the habitat and show that there exists an optimal size reduction.     

Allee effects and extinction in a lattice model

In the interest of conservation, the importance of having a large habitat available for a species is widely known. Here, we introduce a lattice-based model for a population and look at the importance of fluctuations as well as that of the population density, particularly with respect to Allee effects. We examine the model analytically and by Monte Carlo simulations and find that, while the size of the habitat is important, there exists a critical population density below which the probability of extinction is greatly increased. This has large consequences with respect to conservation, especially in the design of habitats and for populations whose density has become small. In particular, we find that the probability of survival for small populations can be increased by a reduction in the size of the habitat and show that there exists an optimal size reduction.     

CXCR4 or CXCR7 antagonists treat endometriosis by reducing bone marrow cell trafficking

Adult stem cells have a major role in endometrial physiology, including remodelling and repair. However, they also have a critical role in the development and progression of endometriosis. Bone marrow‐derived stem cells engraft eutopic endometrium and endometriotic lesions, differentiating to both stromal and epithelial cell fates. Using a mouse bone marrow transplantation model, we show that bone marrow‐derived cells engrafting endometriosis express CXCR4 and CXCR7. Targeting either receptor by the administration of small molecule receptor antagonists AMD3100 or CCX771, respectively, reduced BM‐derived stem cell recruitment into endometriosis implants. Endometriosis lesion size was decreased compared to vehicle controls after treatment with each antagonist in both an early growth and established lesion treatment model. Endometriosis lesion size was not effected when the local effects of CXCL12 were abrogated using uterine‐specific CXCL12 null mice, suggesting an effect primarily on bone marrow cell migration rather than a direct endometrial effect. Antagonist treatment also decreased hallmarks of endometriosis physiopathology such as pro‐inflammatory cytokine production and vascularization. CXCR4 and CXCR7 antagonists are potential novel, non‐hormonal therapies for endometriosis.     

The amplification model for adaptive mutation: simulations and analysis

It has been proposed that the lac revertants arising under selective conditions in the Cairns experiment do not arise by stress-induced mutagenesis of stationary phase cells as has been previously assumed. Instead, these revertants may arise within growing clones initiated by cells with a preexisting duplication of the weakly functional lac allele used in this experiment. It is proposed that spontaneous stepwise increases in lac copy number (amplification) allow a progressive improvement in growth. Reversion is made more likely primarily by the resultant increase in the number of mutational targets--more cells with more lac copies. The gene amplification model requires no stress-induced variation in the rate or target specificity of mutation and thus does not violate neo-Darwinian theory. However, it does require that a multistep process of amplification, reversion, and amplification segregation be completed within approximately 20 generations of growth. This work examines the proposed amplification model from a theoretical point of view, formalizing it into a mathematical framework and using this to determine what would be required for the process to occur within the specified period. The analysis assumes no stress-induced change in mutation rate and describes only the growth improvement occurring during the process of amplification and subsequent elimination of excess mutant lac copies. The dynamics of the system are described using Monte Carlo simulations and numerical integration of the deterministic equations governing the system. The results imply that the amplification model can account for the behavior of the system using biologically reasonable parameter values and thus can, in principle, explain Cairnsian adaptive mutation.     

Multiscale modeling of lipids and lipid bilayers

A multiscale modeling approach is applied for simulations of lipids and lipid assemblies on mesoscale. First, molecular dynamics simulation of initially disordered system of lipid molecules in water within all-atomic model was carried out. On the next stage, structural data obtained from the molecular dynamics (MD) simulation were used to build a coarse-grained (ten sites) lipid model, with effective interaction potentials computed by the inverse Monte Carlo method. Finally, several simulations of the coarse-grained model on longer length- and time-scale were performed, both within Monte Carlo and molecular dynamics simulations: a periodical sample of lipid molecules ordered in bilayer, a free sheet of such bilayer without periodic boundary conditions, formation of vesicle from a plain membrane, process of self-assembly of lipids randomly dispersed in volume. It was shown that the coarse-grained model, developed exclusively from all-atomic simulation data, reproduces well all the basic features of lipids in water solution.     

Lower confidence bounds for prediction accuracy in high dimensions via AROHIL Monte Carlo

MOTIVATION: Implementation and development of statistical methods for high-dimensional data often require high-dimensional Monte Carlo simulations. Simulations are used to assess performance, evaluate robustness, and in some cases for implementation of algorithms. But simulation in high dimensions is often very complex, cumbersome and slow. As a result, performance evaluations are often limited, robustness minimally investigated and dissemination impeded by implementation challenges. This article presents a method for converting complex, slow high-dimensional Monte Carlo simulations into simpler, faster lower dimensional simulations. RESULTS: We implement the method by converting a previous Monte Carlo algorithm into this novel Monte Carlo, which we call AROHIL Monte Carlo. AROHIL Monte Carlo is shown to exactly or closely match pure Monte Carlo results in a number of examples. It is shown that computing time can be reduced by several orders of magnitude. The confidence bound method implemented using AROHIL outperforms the pure Monte Carlo method. Finally, the utility of the method is shown by application to a number of real microarray datasets.     

Multilocus model of sympatric speciation. III. Computer simulations

In the previous papers some analytical results were obtained for the limit stages of sympatric speciation. The present paper aims at finding the scope of validity for these results. The Monte Carlo computer model of this process was created and studied. We deal with two aspects of the speciation process: the development of reproductive isolation between the forming species and the extinction of the intermediate individuals. It has been shown that the advantage of the allele of reproductive isolation increases with the growth of its frequency. The extinction of the intermediates goes differently with various numbers of loci involved in speciation. If reproductive isolation is due to differences in two or four loci, then the completion of extinction of the intermediates requires the strongest disruptive selection, so that the necessary conditions for speciation found previously are also proved to be sufficient. But with eight and probably with a larger number of loci, the selection required to promote speciation in a population that is far from it is considerably stronger than selection needed at the last stage of speciation. Consequently, under some intensities of disruptive selection the final state of a population depends initial state. The conditions under which the stationary state of a population is characterized by bimodal distribution of phenotypes are also found.     

A Machine Learning Method for the Prediction of Receptor Activation in the Simulation of Synapses

Chemical synaptic transmission involves the release of a neurotransmitter that diffuses in the extracellular space and interacts with specific receptors located on the postsynaptic membrane. Computer simulation approaches provide fundamental tools for exploring various aspects of the synaptic transmission under different conditions. In particular, Monte Carlo methods can track the stochastic movements of neurotransmitter molecules and their interactions with other discrete molecules, the receptors. However, these methods are computationally expensive, even when used with simplified models, preventing their use in large-scale and multi-scale simulations of complex neuronal systems that may involve large numbers of synaptic connections. We have developed a machine-learning based method that can accurately predict relevant aspects of the behavior of synapses, such as the percentage of open synaptic receptors as a function of time since the release of the neurotransmitter, with considerably lower computational cost compared with the conventional Monte Carlo alternative. The method is designed to learn patterns and general principles from a corpus of previously generated Monte Carlo simulations of synapses covering a wide range of structural and functional characteristics. These patterns are later used as a predictive model of the behavior of synapses under different conditions without the need for additional computationally expensive Monte Carlo simulations. This is performed in five stages: data sampling, fold creation, machine learning, validation and curve fitting. The resulting procedure is accurate, automatic, and it is general enough to predict synapse behavior under experimental conditions that are different to the ones it has been trained on. Since our method efficiently reproduces the results that can be obtained with Monte Carlo simulations at a considerably lower computational cost, it is suitable for the simulation of high numbers of synapses and it is therefore an excellent tool for multi-scale simulations.     

Effects of radiation quality and oxygen on clustered DNA lesions and cell death

Radiation quality and cellular oxygen concentration have a substantial impact on DNA damage, reproductive cell death and, ultimately, the potential efficacy of radiation therapy for the treatment of cancer. To better understand and quantify the effects of radiation quality and oxygen on the induction of clustered DNA lesions, we have now extended the Monte Carlo Damage Simulation (MCDS) to account for reductions in the initial lesion yield arising from enhanced chemical repair of DNA radicals under hypoxic conditions. The kinetic energy range and types of particles considered in the MCDS have also been expanded to include charged particles up to and including (56)Fe ions. The induction of individual and clustered DNA lesions for arbitrary mixtures of different types of radiation can now be directly simulated. For low-linear energy transfer (LET) radiations, cells irradiated under normoxic conditions sustain about 2.9 times as many double-strand breaks (DSBs) as cells irradiated under anoxic conditions. New experiments performed by us demonstrate similar trends in the yields of non-DSB (Fpg and Endo III) clusters in HeLa cells irradiated by γ rays under aerobic and hypoxic conditions. The good agreement among measured and predicted DSBs, Fpg and Endo III cluster yields suggests that, for the first time, it may be possible to determine nucleotide-level maps of the multitude of different types of clustered DNA lesions formed in cells under reduced oxygen conditions. As particle LET increases, the MCDS predicts that the ratio of DSBs formed under normoxic to hypoxic conditions by the same type of radiation decreases monotonically toward unity. However, the relative biological effectiveness (RBE) of higher-LET radiations compared to (60)Co γ rays (0.24 keV/μm) tends to increase with decreasing oxygen concentration. The predicted RBE of a 1 MeV proton (26.9 keV/μm) relative to (60)Co γ rays for DSB induction increases from 1.9 to 2.3 as oxygen concentration decreases from 100% to 0%. For a 12 MeV (12)C ion (681 keV/μm), the 'predicted RBE for DSB induction increases from 3.4 (100% O(2)) to 9.8 (0% O(2)). Estimates of linear-quadratic (LQ) cell survival model parameters (α and β) are closely correlated to the Monte Carlo-predicted trends in DSB induction for a wide range of particle types, energies and oxygen concentrations. The analysis suggests α is, as a first approximation, proportional to the initial number of DSBs per cell, and β is proportional to the square of the initial number of DSBs per cell. Although the reported studies provide some evidence supporting the hypothesis that DSBs are a biologically critical form of clustered DNA lesion, the induction of Fpg and Endo III clusters in HeLa cells irradiated by γ rays exhibits similar trends with oxygen concentration. Other types of non-DSB cluster may still play an important role in reproductive cell death. The MCDS captures many of the essential trends in the formation of clustered DNA lesions by ionizing radiation and provides useful information to probe the multiscale effects and interactions of ionizing radiation in cells and tissues. Information from Monte Carlo simulations of cluster induction may also prove useful for efforts to better exploit radiation quality and reduce the impact of tumor hypoxia in proton and carbon-ion radiation therapy.