How to give a biological interpretation to this phase portrait?

How to give a biological interpretation to this phase portrait?

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Consider the following system and analyze its behavior.

$$egin{array}{rl} frac{dA}{dt} &= A left( 2-frac{A}{5000}-frac{L}{100} ight) frac{dL}{dt} &= L left(-frac{1}{2}+frac{A}{10000} ight)end{array}$$

The analysis

It has $3$ equilibrium points. I know the stability of the three points but I am not sure if I interpret the meaning of them correctly according to the phase portrait. $x_1, x_2$ are saddle points and $x_3$ is stable point they are

$$overline x_1=(0,0)$$

$$overline x_2=(10000,0)$$

$$overline x=(5000,100)$$

According to the phase portrait I think the behavior of the system is described as:

For every point $(A,L)$ given in the first quadrant, where $A$ is the number of aphids at time $t$ and $L$ is the number of lady-bugs at time $t$, we'll have that in the future, the maximum number of lady-bugs and aphids it's going to be $(5000,100)$ respectively. This also means that both population will never going to extinct.

Even in the case where there were just a few number (near 0) of lady-bugs, their population will grow and will be establish also.

My question

Did I miss something important in the biological description to the phase portrait?

I don't think you missed anything important!

You could investigate the cyclic behaviour around the equilibrium. For example, looking at the variable aphid population size can start above its equilibrium point, then overshoot it and overshoot it again to finally reach the equilibrium point. If you zoom close to the equilibrium point, you might see long cyclic behaviour before ever reaching the equilibrium.

You sure could do all kind of further analyses such as a stability analysis or searching for cyclical equilibrium but it is easy to see with this simple model that nothing very complicated will arise. You could also investigate for what parameter values this classic pattern breaks up if it does.

Another option is to make the model more complex by introducing a carrying capacity (and then investigate under what aphid growth rate the behaviour becomes chaotic), or population structure, or whatever.

You might learn much from it but you might still want to have a look at the post What prevents predator overpopulation?.

I highly recommend the book A Biologist's Guide to Mathematical Modeling in Ecology and Evolution by Otto and Troy. It will show the typical kind of analyses that can be done for this kind of systems.

How to give a biological interpretation to this phase portrait? - Biology

Math 122 - Calculus for Biology II
Fall Semester, 2004
Qualitative Analysis of Differential Equations

© 2000, All Rights Reserved, SDSU & Joseph M. Mahaffy
San Diego State University -- This page last updated 30-Apr-03

Qualitative Analysis of Differential Equations

This section examines nonlinear differential equations using qualitative analysis techniques. Finding analytical solutions to differential equations can be very difficult or impossible, yet often the behavior of the equations can be examined from a qualitative perspective to determine types of behavior, which can lead to insight into modeling problems. In the Math 121 notes, we found that studying equilibria of discrete dynamical systems allowed us to understand much about the behavior of those models. Similarly, qualitative analysis of differential equations can provide valuable insight to complicated biological models.

We begin with the classic example of the logistic growth model, using data from an experiment on bacteria. Equilibria for the model are found and analyzed. A discussion of these techniques is given.

Bacteria grown in a laboratory are innoculated into a broth, then grown for a period of time. Typically, the growth pattern of any bacterial culture follows a regular pattern. The culture has a lag period as the innoculated culture adjusts to the new growing conditions. This is followed by a period of exponential growth , where the bacterial population satisfies the Malthusian growth model. As the population of bacteria increases, certain nutrients become limiting (or there is a build up of waste products that limit growth). The bacterial cell growth slows, and the culture enters a phase called stationary growth . This is characterized by the population leveling off with the culture using different pathways to survive the lean times.

In the previous semester, we studied the discrete logistic growth model that showed this type of behavior under certain conditions on the parameters. However, the discrete logistic growth model could only be simulated and not solved explicitly. Growth of bacteria is a much more continuous process, so their growth is better characterized by a differential equation. In this section, we will develop the continuous logistic growth model for growth of bacteria and show several techniques for analyzing the model and relating it to experimental data.

Professor Anca Segall in the Department of Biology at San Diego State University has done many experiments on the bacterium Staphylococcus aureus , a fairly common pathogen that can cause food poisoning. Standard growth cultures of this bacterium satisfy the classical logistic growth pattern discussed above. Below we present the data from one experiment in her laboratory (by Carl Gunderson), where a normal strain is grown using control conditions and the optical density (OD 650 ) is measured to determine an estimate of the number of bacteria in the culture.

Below is a graph of the data (showing a stained culture of Staphylococcus aureus ) and the logistic growth model that fits the data. It is clear that the model reasonably well matches the actual biological data.

In this experiment, we see that the population of the bacteria begins by growing quite rapidly, but then levels off. Thus, a model for the population growth should have a component corresponding to Malthusian growth and a term accounting for the crowding effects when the nutrient supply has diminished. Recall that the continuous Malthusian growth model was given by the simple differential equation

P ' = kP with P (0) = P 0 , which has the solution P ( t ) = P 0 e -kt .

In our earlier studies, we saw that the discrete logistic growth model was given by

P n+ 1 = P n + rP n (1 - P n / M ) , which can be written P n+ 1 - P n = rP n (1 - P n / M ) .

The term rP n corresponds to the Malthusian growth, while the term - r P n 2 / M is a quadratic term accounting for the crowding effects. As in the arguments made for going from the discrete Malthusian growth model to the continuous Malthusian growth model, where the time interval between n and n +1 becomes small, the continuous logistic growth model follows from the equation above and can be written

This differential equation was first introduced by the Belgian biologist Verhulst ΐ], so it is sometimes referred to as the Verhulst equation . A nonlinear least squares fit to Anca Segall's data in the graph above gives the specific logistic growth model

(Finding the parameters for this system uses a nonlinear least squares routine. To learn more about fitting the parameters to the logistic growth model, lecture notes are available from my website.)

Can we solve this differential equation?

There is a solution to the logistic growth differential equation, which can be found in a hyperlink to this section (Solution to the Logistic Growth Model). In fact, there are a couple of methods that can solve this differential equation, either separation of variables (which then uses special integration techniques) or Bernoulli's method. However, since many differential equations cannot be solved or involve complicated methods, we want to develop simpler techniques to understand the qualitative behavior of differential equations .

Qualitative Behavior of Differential Equations

The continuous logistic growth model can be analyzed without finding the exact solution. The techniques follow a similar analysis to the discrete dynamical systems. We first examine the differential equation to find the equilibria , then we determine the behavior of the solutions near those equilibria. Below we detail a typical analysis using the logistic growth equation.

Consider the logistic growth equation:

We begin by drawing a graph of the function on the right hand side of the differential equation, f ( P ) .

The graph of f ( P ) shows a parabola that intersects the P -axis at P = 0 and P = 2000 . These P -intercepts are where f ( P ) = 0 or

This means that there is no change in the growth of the population or the population is at equilibrium . This is the first step in any qualitative analysis of a differential equation. Find all equilibria for the differential equation, which are simply all points where the derivative of the unknown function is zero.

The graph of f ( P ) gives us more information about the behavior of the logistic growth model. As noted above, the equilibria occur at P e = 0 and P e = 2000 . Notice that to the left of P e = 0 , f ( P ) < 0 , while to the right of P e = 0 , f ( P ) > 0 . Thus, when P < 0 , dP/dt < 0 and P is decreasing. (Note that this region is outside the region of biological significance.) When 0 < P < 2000 , then dP/dt > 0 and P is increasing. Furthermore, we can see that the greatest increase in the growth of the population occurs at P = 1000 , where the vertex of the parabola occurs.When P > 2000 , then again dP/dt < 0 and P is decreasing.

This information allows us to draw what is called a Phase Portrait of the behavior of this differential equation along the P -axis. The behavior of the differential equation is denoted by arrows along the P -axis, which are easily directed by whether the function f ( P ) is positive or negative. We use a solid dot to represent an equilibrium, which solutions of the differential equation approach asymptotically, and an open dot represents an equilibrium, where solutions nearby move away from the equilibrium. This is diagrammed below using the graph of f ( P ) .

We can easily sketch the behavior of the solutions as functions of time using this phase portrait. Below is a graph of the solutions P ( t ) with t being the independent variable. The equilibria give rise to horizontal lines (meaning that P ( t ) does not change with time). When the initial values of P have the phase portrait pointing to the left, then the solution is decreasing, while arrows to the right on the phase portrait have the solutions increasing. The solutions are not allowed to cross paths in the time-varying diagram shown below.

The solutions above show that all positive initial conditions result in solutions eventually tending towards the equilibirum at 2000 . The slope of the solution at any time can be seen in the figure above with the slope matching the value of f ( P ) .

There are more Worked Examples through the usual hyperlink.

Stephen Jay Gould wrote many articles for Natural History on various issues of evolution and was a leading voice against the teaching of "Creation Science." He was also an expert on the evolution of gastropods. The shell of a snail exhibits chirality, which means that the snail shell develops either a left-handed ( sinistral ) or right-handed ( dextral ) coil relative to the central axis. In Α], he writes that the Indian conch shell, Turbinella pyrum , is primarily a right-handed gastropod. However, left-handed shells exist and are "exceedingly rare." (The Indians view the shells as right-handed from their perspective, which makes the rare shells very holy.) The Hindu god "Vishnu, in the form of his most celebrated avatar, Krishna, blows this sacred conch shell to call the army of Arjuna into battle." So why does nature favor snails with one particular handedness? Gould notes that the vast majority of snails grow the dextral form.

One mathematical model discussed in the book by Clifford Henry Taubes Β] gives a simple mathematical model to predict the bias of either the dextral or sinistral forms for a given species.

  • Assume that the probability of a dextral snail breeding with a sinistral snail is proportional to the product of the number of dextral snails times sinistral snails.
  • Assume that two sinistral snails always produce a sinistral snail and two dextral snails produce a dextral snail. Assume that a dextral-sinistral pair produce dextral and sinistral offspring with equal probability.

One could and should debate these assumptions. Note that the first assumption implicitly assumes that given a choice of mates a dextral snail is twice as likely to choose a dextral snail than a sinistral snail. (You can check this implication using basic probability.) It would be useful to have real experimental verification of these assumptions.

Let p ( t ) be the probability that a snail is dextral. Taubes presents the following model that qualitatively exhibits the behavior described by the assumptions above.

where a is some positive constant. We would like to see the behavior of this differential equation and determine what its solutions predict about the chirality of populations of snails.

Though the differential equation above could be solved using separation of varibles and some integration techniques, the solution is not particularly easy to obtain. However, by using the geometric approach afforded by our qualitative analysis techniques, this differential equation can be analyzed relatively easily to determine why snails are likely to be in either the dextral or sinistral forms.

The analysis begins by graphing the function of p on the right hand side of the differential equation. From the graph below, we see that the p -intercepts are 0 , 1/2 , and 1 . Thus, the equilibria for this model are p e = 0 , 1/2 , and 1 . In addition, we see that the function is negative for 0 < p < 1/2 and positive for 1/2 < p < 1 . It follows that all solutions of the differential equation decrease whenever 0 < p < 1/2 and increase whenever 1/2 < p < 1 . Thus, we have that the equilibrium at p e = 1/2 is unstable (denoted by the open circle), and the equilibria at p e = 0 and 1 are stable (denoted by the closed circles). The phase portrait of this model is shown on the p -axis with the graph of the function on the right hand side of the differential equation drawn to indicate the direction the arrows should be pointing.

Below we show the actual solutions as drawn by Maple for a collection of different initial conditions using the solution space with t as the independent variable and p as the dependent variable. These solutions are shown below.

From both of the diagrams above it is easy to see that for this model, the solutions tend toward one of the stable equilibria, p e = 0 or 1 . In the case that the solution tends toward p e = 0 , then the probability of a dextral snail being found drops to zero, so the population of snails all have the sinistral form. When the solution tends toward the equilibrium p e = 1 , then the population of snails virtually all have the dextral form. This is what is observed in nature suggesting that our model exhibits the behavior of the evolution of snails. This does not mean that the model is a good model! It just means that the model exhibits the basic behavior observed experimentally from the biological experiments.

Ώ] T. Carlson Über Geschwindigkeit und Grösse der Hefevermehrung in Würze. Biochem. Z. 57: 313-334, 1913.

ΐ] P. F. Verhulst, "Notice sur la loi que la population suit dans son accroissement," Corr. Math. et Phys. 10 (1838), 113-120.

Α] S. J. Gould, "Left Snails and Right Minds," Natural History , April 1995, 10-18, and in the compilation "Dinosaur in a Haystack" ( 1996)

Β] C. H. Taubes, Modeling Differential Equations in Biology , Prentice Hall, 2001.

Mathematical modeling of virus-cell interaction has a long history. Grounded in the vast and diverse theoretical epidemiology field, these mathematical models serve as valuable tools to explain empirical data, predict possible outcomes of virus infection, and propose the optimal strategy of anti-virus therapy [1, 2]. The unquestionable success of mathematical models of certain virus-host systems, in particular, HIV infection [3, 4] provides for reasonable hope that substantial progress can be achieved in other areas of virology as well.

Equally extensive efforts have been dedicated over many years to mathematical modeling of cancer development. Stochastic models that take into account random mutations and cell proliferation proved to be useful in the context of epidemiology and statistical data [5] and for modeling cancer initiation and progression in terms of somatic evolution [6]. Deterministic models of tumor growth have proved valuable as well. Many of these have addressed avascular and vascular tumor growth taking advantage of methods borrowed from physics [7, 8] but some use population ecology models to treat tumor as a dynamic society of interacting cells [9–11]. A variety of mathematical approaches contribute to modeling cancer progression from different standpoints and take stock of various factors affecting tumor growth ([12, 13] and references therein).

Here we address a complex process that involves both virus-cell interaction and tumor growth, namely, the interaction of the so-called oncolytic viruses with tumors. Oncolytic viruses are viruses that specifically infect and kill cancer cells but not normal cells [14–17]. Many types of oncolytic viruses have been studied as candidate therapeutic agents including adenoviruses, herpesviruses, reoviruses, paramyxoviruses, retroviruses, and others [15, 17]. Probably, the best-characterized oncolytic virus, that has drawn a lot of attention, is ONYX-015, an attenuated adenovirus that selectively infects tumor cells with a defect in the p53 gene [16]. This virus has been shown to possess significant antitumor activity and has proven relatively effective at reducing or eliminating tumors in clinical trials [18–20]. Thus, in studies of patients with squamous cell carcinoma of the head and neck the response rate was significantly higher (78%) in patients who received the combination of viral therapy and chemotherapy than in patients who were treated with chemotherapy alone (39%). Furthermore, a small number of patients who were treated with the oncolytic virus showed regression of metastases [15]. Although safety and efficacy remain substantial concerns, several other oncolytic viruses acting on different principles, including tumor-specific transcription of the viral genome, have been developed, and some of these viruses have entered or are about to enter clinical trials [15, 21–23].

The oncolytic effect can result from at least three distinct modes of virus-host interaction [15, 17]. The first mode involves repeated cycles of viral replication in the tumor cells leading to cell death and, consequently, to tumor reduction and, potentially, elimination. The second mode consists in low-level virus reproduction that, however, results in the production of a cytotoxic protein, which then causes cell damage. The third mode involves virus infection of cancer cells that induces antitumor immunity. Cancer cells possess weak antigens for host immune sensitization. Virus infection causes inflammation and lymphocyte penetration into the tumor, with the virus antigens eliciting increased sensitivity to tumor necrosis factor-mediated killing.

Although the indirect modes of virus cancer therapy based on production of cytotoxic proteins or antitumor immunity may be promising, direct lysis of tumor cells by an oncolytic virus is the current mainstream strategy. The interactions between the growing tumor and the replicating oncolytic virus are highly complex and nonlinear. Thus, to precisely define the conditions that are required for successful therapy by this approach, mathematical models are needed. Experiments on human tumor xenografts in nude mice have shown that the effect of oncolytic virus infection on tumors can range from no apparent effect, to reduction and stabilization of the tumor load (i.e., the overall size of a tumor), to elimination of the tumor [24]. Complete regression of tumors has been reported also in some patients treated with oncolytic viruses as part of clinical trials [25]. However, the simplest mathematical models describing a growing tumor infected with an oncolytic virus fail to incorporate all possible outcomes in particular, these models do not allow tumor elimination [12, 26]. Here, we present a conceptual model of tumor cells-virus interaction, which, depending on system parameter values, exhibits various behaviors including deterministic elimination of the cancer cells.

Several mathematical models that describe the evolution of tumors under viral injection were recently developed. Our model builds upon the model of Wodarz [12, 26] but introduces several plausible modifications. Wodarz [12, 26] presented a mathematical model that describes interaction between two types of tumor cells (the cells that are infected by the virus and the cells that are not infected but are susceptible to the virus so far as they have the cancer phenotype) and the immune system. Here, we consider only direct killing of tumor cell by an oncolytic virus and, accordingly, disregard the effects of the immune system. The resulting model has the general form

where X(t) and Y(t) are the sizes of uninfected and infected cell populations, respectively f i(X, Y) i = 1, 2, are the per capita birth rates of uninfected and infected cells and g(X, Y) is a function that describes the force of infection, i.e., the number of cells newly infected by the virus released by an infected cell per time unit. Note that there is no separate equation for free virions it is assumed that virion abundance is proportional to infected cell abundance, which can be justified if free virus dynamics is fast compared to infected cell turnover [2]. The model also assumes that, upon division of infected cells, the virus is passed on to both daughter cells. This is certain for viruses that integrate into the tumor cell genome but this assumption should also be appropriate for non-integrating viruses because active virion production should result in a very high probability that the virus is transmitted to both daughter cells. The functions used by Wodarz [26] are

where r1, r2, d, a, b, K are non-negative parameters. The assumptions are that the tumor grows in a logistic fashion (possibly, with different rates of growth for the uninfected and infected tumor cells), and the incidence of infection is proportional to the product XY the latter assumption is based on an analogy with chemical kinetics, namely, the law of mass action.

The main result of the analysis of model (1)-(2) consists in defining conditions required for maximum reduction of the tumor load. It has been suggested that "because we used deterministic model, the tumor can never go completely extinct but can be reduced to very low levels" elimination of the tumor then might occur through stochastic effects which are not part of the model per se [26]. In contrast, here we show that a straightforward modification of model (1)-(2) can lead to dynamical regimes that describe deterministic elimination of the tumor cells.

Other mathematical models for tumor-virus dynamics are, mainly, spatially explicit models, described by systems of partial differential equations (PDE) (which is an obvious and necessary extension of ODE models inasmuch as most solid tumors have distinct spatial structure) the local dynamics, however, is usually modeled by systems of ODE that bear close resemblance to a basic model of virus dynamics [1]. Wu et al. modeled and compared the evolution of a tumor under different initial conditions [27]. Friedman and Tao (2003) presented a rigorous mathematical analysis of a somewhat different model [28]. The use of PDE for the virus spread is the main feature that distinguishes the model of Friedman and Tao [28] from the model of Wu et al. [27]. Recently, Wein et al. [29] incorporated immune response into their previous model [27]. In this new study, recent preclinical and clinical data were used to validate the model and estimate the values of several key parameters, and it has been concluded that oncolytic viruses should be designed for rapid intratumoral spread and immune avoidance, in addition to tumor-selectivity and safety [29]. In a more recent study, an analysis of an ODE system, which is a simplified approximation to the previously described PDE model and bears some similarities to the model of Wodarz, has been described [30]. Tao and Guo [31] recently extended the model of Wein et al. [29], proved global existence and uniqueness of solution for the new model, studied the dynamics of this novel cancer therapy, and explored an explicit threshold of the intensity of the immune response for controlling the tumor. Wodarz also developed an extension of his previous model to study advantages and disadvantages of replicating versus nonreplicating viruses for cancer therapy [32].

A distinct aspect of all these models is the description of the process of infection (or, if free virus dynamics is explicitly modeled, the virus-cell contact as well) using the law of mass action, which states that the rate of change of the uninfected cell population is proportional (if no demography effects are taken into account) to the product XY (where X and Y are as before, or Y stands for virus population if the latter is included into the model). Under mass-action kinetics and the assumption of infinitesimally short duration of contact and homogeneous mixing of the cell populations, the contact rate is proportional to the product XY of the respective densities. There are situations when mass action can be a good approximation however, in many real-life cases, it is only acceptable when X

Y, giving unrealistic rates when X >> Y or X <<Y. In particular, for large populations of cells, finite and often slow spread of the virus prevents it from infecting a large number of cells per infected cell per unit of time such that a more realistic approximation of the infection process is required. The assumption underlying mass action is that the contact rate is a linear function of density, N = X + Y. At the other extreme, the contact rate might be independent of host density. Assuming that infected and uninfected hosts are randomly mixed, this would lead to transmission function of the form bXY/(X + Y). This mode of transmission is often called 'frequency-dependent' transmission [33]. This transmission function makes sense when X >> Y or X <<Y because both extremely low and extremely high rates of transmission are excluded from the consideration. The mass action assumption, which goes back to the pioneering work of Kermak and McKendrick [34], has almost always been used for transmission in host-pathogen models, in some cases, non-critically. Other modes of transmission have been used [33], and, importantly, can yield quite diverse results. We emphasize that the mode of transmission determines probable responses of the system to control, so it is vital to identify the most appropriate approach to model transmission. In particular, mathematically, 'frequency-dependent' transmission, because of the non-analytical vector field at the origin, yields a qualitatively different outcome, compared to 'mass-action' transmission.

The model (1)-(2) is a version of the classical predator-prey model of a biological community first developed by Lotka [35] and Volterra [36] in 1925–1931 the term bXY describes the simplest correspondence between prey consumption and predator production similar to the law of mass action. A crucial element in models of biological communities in the form (1) is the functional response g(X, Y), i.e., the number of prey consumed per predator per time unit for the given numbers of prey X and predators Y. In the Volterra model and in model (1)-(2), this function is bX. Another well-known model has been developed by Holling [37] and has been widely applied in epidemiology [38]. Under this model, g(X, Y) = bX/(1 + abX), which takes into account the saturation effect. These two types of possible functional responses (and many others) do not depend on predator density, i.e., g(X, Y) = g(X), and, accordingly, have been named 'prey-dependent' by Arditi and Ginzburg [39]. In many cases, it is more realistic to assume that the functional response is ratio-dependent (g(X, Y) = g(z), where z = X/Y [39]). If we consider a Holling-type function g(z) = bz/(1 + z), then we again obtain

In (3), the meaning of b is the infection rate, i.e., the mean number of infections an infected cell can cause per unit of time. In the terminology of epidemic models, such a rate term would be said to reflect proportional mixing as opposed to homogeneous mixing [40].

The ratio-dependent models present a challenge with regard to their dynamics near the origin due to the fact that they are undefined at (0, 0). Berezovskaya et al. showed that, depending on parameter values, the origin can have its own basin of attraction in the phase space [41], which corresponds to the deterministic extinction of both species [40–43]. In the context of the interaction between oncolytic viruses and tumors, it is clear that the ratio-dependent models display original dynamic properties that could be directly relevant for the possibility of tumor eradication by virus therapy.

Here, we show that a plausible change of the dynamical system modeling the growth of two competing populations of cells, one of which is infected by a virus and the other one is not infected, can result in a remarkable change in the model dynamics. Moreover, the additional dynamical regimes, which do not emerge in the original model, might be particularly important with respect to the underlying biological problem, the oncolytic virus therapy for cancers.

Model and methods

To develop our methodology for analysing transient behaviour in non-autonomous dynamical systems, we use a simple toggle switch model (see [22], and references therein) with time-dependent parameters. We consider two interacting genes X and Y (Figure 1B, panel 1). Concentrations of the corresponding protein products are labelled x and y. X and Y mutually repress each other, are linearly activated by external signals and can auto-activate themselves (Figure 1B, panel 1). Protein products decay linearly dependent on their concentration. The mathematical formulation of our toggle switch model is thus given by

where parameters α x and α y represent the external activation on genes X and Y respectively. Sigmoid functions with Hill coefficients of 4 are used to represent auto-activation and mutual repression, where parameters a and c determine auto-activation thresholds, while b and d determine thresholds for mutual repression. Protein decay rates are represented by parameters λ x and λ y.

The toggle switch model (1) exhibits different dynamical regimes depending on the values of its parameters (Figure 2A–C). Its name derives from the fact that it can exhibit bistability over a wide range of parameters. When in this bistable region of parameter space, the underlying phase portrait has two attracting states and one saddle point (Figure 2B). All phase portraits associated with parameters in the bistable range are topologically equivalent to each other, meaning that they can be mapped onto each other by a continuous deformation of phase space called a homeomorphism [34].

Dynamical regimes of the toggle switch model. The toggle switch model can exhibit three different dynamical regimes depending on parameter values. (A) In the monostable regime, the phase portrait has one attractor point only (represented by the blue dot on the quasi-potential landscape). At this attractor, both products of X and Y are present at low concentrations. (B) In the bistable regime, which gives the toggle switch its name, there are two attractor points (shown in different shades of blue) and one saddle (red) on a separatrix (black line), which separates the two basins of attraction. The attractors correspond to high x, low y (dark blue), or low x, high y (light blue). The two factors never coexist when equilibrium is reached in this regime. (C) In the tristable regime, both bistable switch attractors and the steady state at low co-existing concentrations are present (shown in different shades of blue). In addition, there are two separatrices with associated saddle points (red). These regimes convert into each other as follows (double-headed black arrows indicate reversibility of bifurcations): the monostable attractor is converted into two bistable attractors and a saddle point through a supercritical pitchfork bifurcation the saddle in the bistable regime is converted into an attractor and two additional saddles in the tristable regime through a subcritical pitchfork bifurcation the bistable attractors and their saddles collide and annihilate in two simultaneous saddle-node (or fold) bifurcations to turn the tristable regime into a monostable one. Graph axes as in Figure 1B, Panel 4.

The toggle switch model has two other dynamical regimes: monostable and tristable. Phase portraits associated with parameters in the monostable range have only one attractor point (Figure 2A), while those in the tristable range have three attractor states and two saddle points (Figure 2C). Again, phase portraits within each regime are topologically equivalent to each other. While phase space can be geometrically deformed within each regime (through movements of attractors or separatrices), its topology only changes when one regime transitions into another through different types of bifurcations [32, 34] (Figure 2). The transition from monostable to bistable is known to be governed by a supracritical pitchfork, the transition from bistable to tristable involves a subcritical pitchfork bifurcation, and the transition from tristable to monostable takes place through two simultaneous saddle-node bifurcations involving the two attractors labelled in darker blue in Figure 2C.

Definition of the potential landscape

Potential landscapes can only be calculated explicitly for the class of dynamical systems called gradient systems [32]. A two-variable gradient system is a dynamical system

which satisfies the following relationship between partial derivatives

For gradient systems, it is possible to calculate a closed-form (explicit) potential function, V(x,y) such that

The local minima on the two-dimensional potential surface given by V(x,y) correspond mathematically to the steady states of the system in (2) since, if (x ∗ , y ∗ ) is such that

and, therefore, (x ∗ , y ∗ ) is a steady state of (2).

Calculating quasi-potential landscapes

Condition (3) will not always be met. In particular, dynamical systems representing gene interaction networks are not in general gradient systems, and therefore an associated potential function and landscape may not exist. In such cases, we can still take advantage of the visualisation power of potential landscapes by approximating the true potential using a numerical method. The numerical approximation method we adopt for our study was developed by Bhattacharya and colleagues [38] using a toggle switch model very similar to the one used here. This allows us to calculate a quasi-potential landscape for any specific set of fixed parameter values.

The quasi-potential, which we denote by V q, is defined to decrease along all trajectories of the dynamical system as they progress on the phase portrait over time

Δ x and Δ y are defined as small-enough increments along the trajectory such that d x/d t and d y/d t can be considered constant in the interval [(x,x+Δ x),(y,y+Δ y)]. In addition, Δx = dx dt Δt and Δy = dy dt Δt , where Δ t is the time increment. Substituting into equation 7, we obtain

Δ V q has been formulated in such a way that, for positive time increments Δ t, Δ V q is always negative along the unfolding trajectory and is, in effect, a Lyapunov function of the two-gene dynamical system [32]. This ensures that trajectories will always “roll” downhill on the quasi-potential surface. Just as in the case of closed-form potential, the steady states of the system (x ∗ ,y ∗ ) correspond to the local minima on the quasi-potential surface since Δ V q(x ∗ ,y ∗ )=0.

We apply the numerical approximation method described above to trajectories with various initial points on the x-y plane. This yields a sampled collection of trajectories with quasi-potential values associated to every one of their points. Next, we apply the following two assumptions, in order to construct a continuous quasi-potential surface from this sample of discrete trajectories [38]:

Two trajectories with different initial conditions that converge to the same steady state must also converge to the same final quasi-potential level (normalisation within basins of attraction).

Two adjacent trajectories that converge to different steady states will be taken to start from the same initial quasi-potential level (normalisation between basins of attraction).

Finally, interpolation of all the normalised trajectories results in a continuous quasi-potential landscape. Bhattacharya et al.[38] validated this approach by demonstrating that the quasi-potential values of the steady states were inversely correlated with their probability of occurrence using a stochastic version of the toggle switch dynamical system.

Approximating non-autonomous trajectories

As we have argued in the Background Section, we cannot generally assume that parameter values remain constant over time when modelling biological processes. We take a step-wise approximation approach to the change in parameter values to address this problem (Figure 3). We chose a time increment (step size) as small as possible. Parameter values are kept constant for the duration of each time step. As a consequence, the associated phase portrait will also remain constant during this time interval, and is visualised for each step by calculating a quasi-potential landscape as described in the previous section (Figure 3C, top row).

Numerical approximation of non-autonomous trajectories. (A) Toggle switch network. Red arrows representing auto-activation indicate time-dependence of threshold parameters a x and a y (see equation 1). (B) Values of auto-activation thresholds a x and a y are altered simultaneously and linearly over time. The graph shows the step-wise approximation of a continuous change, in this case, an increase in parameter values. Step size is taken as small as computational efficiency allows. (C) During every time step, parameters can be considered constant, and the phase portrait and (quasi-)potential landscape are calculated for the current set of parameter values. Trajectories are then integrated over the duration of the time step using the previous end point as the current initial condition. The result is mapped onto the potential surface. The four panels in (C) show examples of potential landscapes (upper panels) calculated based on sets of parameter values at time points indicated by dashed arrows from (B). Important events altering the geometry of the trajectory are indicated. Lower panels show the corresponding instantaneous phase portraits with the integrated progression of the trajectory across time steps. See Model and methods for details.

The smaller the time increments considered, the better we are able to approximate continuous changes in parameter values, and the consequent changes to the associated phase portrait and quasi-potential landscape. Such accurate approximation allows us to faithfully reproduce non-autonomous trajectories produced by models with continuously time-variable parameters. This is done by integrating trajectories using constant parameters during each time step, and then using the resulting end position in phase space as the initial condition for the next time step. The resulting integrated trajectories can then be visualised by mapping them from the underlying phase plane onto the associated quasi-potential landscape as described above. This allows us to track and analyse in detail how changes in the phase portrait and quasi-potential landscape shape the trajectories as the values of the parameters are changing.

Tuesday, December 29, 2020

Evolution and Evolutionary Algorithms: Selection, Mutation, and Drift

A guide to the threepressures that shape innovation in living and non-living systems.

(a version of this article has also been posted to Medium)

I teach a course on Bio-Inspired AI and Optimization that is meant to be a graduate-level survey of nature-inspired algorithms that also provides a more serious background in the natural-science (primarily biological) fundamentals underlying the inspiration. The first half of the course covers nature-inspired optimization metaheuristics, with a heavy focus on evolutionary algorithms. An evolutionary algorithm is a scheme for automating the process of goal-directed discovery to allow computers to find innovative solutions to complex problems. There is a wide range of evolutionary algorithms, but a common feature of each is that the computer generates a population of random candidate solutions, evaluates the performance of each of these candidates, and then uses the best of these candidates as “parents” to guide the generation of new candidates in the next generation.

Most of my students come from computer science or engineering backgrounds and, as such, have very little formal education in biology let alone something as specific as population genetics (“popgen”). However, to really understand the complex process of evolutionary innovation inherent to evolutionary algorithms (and evolutionary computing in general), it requires at least some fundamental background in popgen. I think when most people reflect back on their high-school biology courses, they might remember something about natural selection and mutation being important in thinking about the evolution of adaptations in natural populations. However, there is a third evolutionary force that is extremely important — especially when considering small populations, like the ones that are artificially generated in an evolutionary algorithm. That force is (genetic) drift. So let’s review all three:

    Natural selection reflects that some individuals in a population will be at a fundamental disadvantage with respect to other individuals. Those individuals (who are, in the computational creativity context, are relatively poor solutions to a problem) will be very likely to be “selected out” in large populations because there will be so many other individuals who are relatively “fitter.” “Fitness” is a measure of how many offspring an individual can put into the next generation given the current context. If some individuals can put more individuals into the next generation than others, they are “more fit.” If all individuals have the same fitness, then every parent has the same chance of getting her offspring into the next generation. If some individuals have less fitness than others, then they have less chance of getting their offspring into the next generation.

Some people are taught that natural selection only matters when resources are scarce and thus population sizes are limited (thus making individuals compete for opportunities). This is not the whole story and is why we must discuss (genetic) drift below. Before getting into that, note that even in populations that are not limited, differences in the rates of growth of different strategies will gradually change the relative share a strategy has of a population. So even without resource limitation, differences in “relative fitness” will naturally select for the most fit individuals to have the strongest share of the population.

Fortunately, mutation (mentioned above) can rescue us from drift. Mutation introduces new variation in a population, and natural selection can choose strategies out of that new variation. So if we want to combat drift, we can just crank up the mutation rate. The downside of that is that the mutation rate also quickly corrupts well-performing strategies. So populations that have a high mutation rate will tend to have a diverse set of strategies within them and maintain a diverse set of fitnesses. Some individuals will have very high fitness, but they will co-exist with individuals with very low fitness (due to mutation) that are just a side effect of the stabilizing force of mutation. Reducing the mutation rate helps to ensure all solutions have similar fitness, but there is never any way to know if a population of individuals with similar fitness is because their shared strategy is good or they simply reached fixation too soon.

So when building an evolutionary algorithm, it is important to start with a diverse population and then build mutation and selection operations that maintain diversity as long as possible (staving off genetic drift). So long as the population is diverse, natural selection will continue to explore large regions of the strategy space. However, if mutation is too strong, then it will limit exploitation and tuning of strategies because adaptations that make small changes in fitness will quickly be lost to mutation. Consequently, if you have the computational budget, it is best to build very large population sizes with very low mutation rates and choose selection operators that moderate selection pressure — giving low-fitness strategies a chance to stay in the large population pool.

Similarly, when thinking about evolution in natural systems, it is important to remember how large the ancestral populations were. Those that evolved in large-population contexts may tend to show more signs of natural selection (and will likely have evolved mechanisms to reduce the mutation rate). Those that evolved in small-population contexts may tend to have high mutation rates and show diversity patterns more closely related to randomness. This latter case relates to neutral theories of evolution, which are important to consider when trying to understand the source of observed variation in systems we see today.

This story is summarized in the graphic I’ve prepared above, which shows mutation and natural selection as forces re-shaping populations within a drift field that, in the absence of those forces, will eventually homogenize the population on an arbitrary strategy.

So how do we come up with interesting new ideas for mutation and selection operators for evolutionary algorithms? We should continue to look at population genetics. In fact, some theories in population genetics (like Sewall Wright’s shifting-balance theory) are much better descriptors of evolutionary algorithm behavior than the more complex evolutionary trajectories of living systems. For example, distributed genetic algorithms, which create islands of evolutionary algorithms that only exchange population members across island boundaries infrequently, tend to out-perform conventional genetic algorithms on the same computational budgets for reasons that make sense in light of population genetics. This is a more advanced topic, and you’re welcome to read/listen more about this in my CSE/IEE 598 lectures. For now, I hope you look at living and non-living populations around you through the lenses of mutation, drift, and natural selection.

Vertebrate Genomes Project Publishes Lessons Learned From 16 Reference Genome Assemblies

In April of this year the Vertebrate Genomes Project (VGP) announced their flagship study including high-quality, near error-free, and near complete reference genome assemblies for 16 species representing six major lineages of vertebrates, including mammals, reptiles, monotremes, amphibians and fish. The lessons learned from these first 16 genome assemblies will help guide the project in their goal of producing reference genomes for the approximately 70,000 living vertebrates. The availability of this genomic data would have wide applications from basic research to conservation.

Harris Lewin, distinguished professor of evolution and ecology in the UC Davis College of Biological Sciences, who serves on the VGP’s leadership council, and postdoctoral researcher Joana Damas are among the coauthors on the Nature paper.

Taking advantage of dramatic improvements in sequencing technology, the study details numerous technological improvements based on these 16 genome assemblies including novel algorithms that put the pieces of the genome puzzle together.

“Completing the first vertebrate reference genome, human, took over 10 years and $3 billion dollars. Thanks to continued research and investment in DNA sequencing technology over the past 20 years, we can now repeat this amazing feat multiple times per day for just a few thousand dollars per genome,” said Adam Phillippy, chair of the VGP genome assembly and informatics working group of over 100 members and head of the Genome Informatics Section of the National Human Genome Research Institute at the NIH in Bethesda, Md.

Comparing bat genomes

The excellent quality of these genome assemblies enables unprecedented novel discoveries which have implications for characterizing biodiversity for all life, conservation, and human health and disease.

Lewin and Damas contributed a comparison of six bat species’ genomes, with Canada lynx, platypus and chicken genomes as outgroups and the human genome as a reference. They were able to define conserved blocks of DNA and evolutionary breakpoints, showing that the rate of evolution accelerated in bats after the last mass extinction 66 million years ago.

“The completeness of these genomes allowed us to look very thoroughly at the breakpoint regions, which flank chromosomal rearrangements,” Damas said.

By doing that, they also found some chromosomal rearrangements associated with gene loss in bats, including one associated with loss of genes related to the immune system.

“We found bat species to have lost from two to the twelve genes present in this locus in humans,” Damas said.

These gene losses could be related to bat-specific differences in immunity to infectious agents including SARS-CoV-2, the virus that causes COVID-19, Lewin said.

Data from the project could also play a role in protecting rare and endangered species. In collaboration with officials in Mexico, genomic analysis of the vaquita, a small porpoise and the most endangered marine mammal, suggest that harmful mutations are purged from these small populations in the wild, giving hope for the species’ survival.

Thousands of species, hundreds of scientists

The VGP involves hundreds of international scientists working together from more than 50 institutions in 12 different countries. As the first large-scale eukaryotic genomes project to produce reference genome assemblies meeting a specific minimum quality standard, the VGP has thus become a working model for other large consortia, including the Bat 1K, Pan Human Genome Project, Earth BioGenome Project, Darwin Tree of Life, and European Reference Genome Atlas, among others. The Earth BioGenome Project is chaired by Lewin, and the secretariat of the project is located on the UC Davis campus.

As a next step, the VGP will work to complete phase one of the project, approximately one representative species from each of 260 vertebrate orders separated by a minimum of 50 million years from a common ancestor.

Phase two will focus on representative species from each vertebrate family and is currently in the progress of sample identification and fundraising.

The project also collaborated with DNAnexus and Amazon to generate a publicly available VGP assembly pipeline and host the genomic data in the Genome Ark database. The genomes, annotations and alignments are also available in international public genome browsing and analyses databases. All data are open source and publicly available under the G10K data use policies.

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Translated by Henri Atlan, see: Atlan [1], p. 508.

A criticism already outlined by Thom and Chumbley in “Stop Chance! Silence Noise!” [20] against the work of Monod [21], Prigogine and Stengers [22], Atlan [23], and Serres [24].

Entropy and noise are equated in Wiener [25]. Wiener established the connection between Shannon’s concept of entropy as a measure of uncertainty in communication with Schrödinger’s concept of negative entropy as a source of order in living organisms. Shannon’s theory looked at the accurate transmission of messages. In this context, the information transmitted is equivalent to the entropy of the information source: the higher the initial uncertainty (entropy), the higher the amount of information achieved in the end. For Shannon, higher entropy indicated more information for Schrödinger, on the other hand, higher entropy indicated less order. Wiener brought into line these approaches. He redefined the notion of entropy, related not to the initial uncertainty (as in Shannon’s definition) but to the degree of uncertainty remaining after the message has been received. Higher entropy (noise) now entailed less information. Thus, while Shannon’s information is equivalent of regular entropy, Wiener’s information is equivalent of negative entropy. Due to this alteration in sign, information (order) now opposes entropy both in information theory and in biological organization. Wiener successfully presented a notion of information (negative entropy) as a general measure of certainty, order and organization in any given “system”, whether living or technical.

Please note that I am not talking of physical invariance/stability.

Such as the two sides of a coin (constraints) enabling to describe (and thus to determine) the toss of a coin.

See as well Châtelet [53], p. 27

Data analysis: A complex and challenging process

Though it may sound straightforward to take 150 years of air temperature data and describe how global climate has changed, the process of analyzing and interpreting those data is actually quite complex. Consider the range of temperatures around the world on any given day in January (see Figure 2): In Johannesburg, South Africa, where it is summer, the air temperature can reach 35° C (95° F), and in Fairbanks, Alaska at that same time of year, it is the middle of winter and air temperatures might be -35° C (-31° F). Now consider that over huge expanses of the ocean, where no consistent measurements are available. One could simply take an average of all of the available measurements for a single day to get a global air temperature average for that day, but that number would not take into account the natural variability within and uneven distribution of those measurements.

Figure 2: Satellite image composite of average air temperatures (in degrees Celsius) across the globe on January 2, 2008 ( image © University of Wisconsin-Madison Space Science and Engineering Center

Defining a single global average temperature requires scientists to make several decisions about how to process all of those data into a meaningful set of numbers. In 1986, climatologists Phil Jones, Tom Wigley, and Peter Wright published one of the first attempts to assess changes in global mean surface air temperature from 1861 to 1984 (Jones, Wigley, & Wright, 1986). The majority of their paper – three out of five pages – describes the processing techniques they used to correct for the problems and inconsistencies in the historical data that would not be related to climate. For example, the authors note:

Early SSTs [sea surface temperatures] were measured using water collected in uninsulated, canvas buckets, while more recent data come either from insulated bucket or cooling water intake measurements, with the latter considered to be 0.3-0.7° C warmer than uninsulated bucket measurements.

0.5° C to early canvas bucket measurements, but it becomes more complicated than that because, the authors continue, the majority of SST data do not include a description of what kind of bucket or system was used.

Similar problems were encountered with marine air temperature data. Historical air temperature measurements over the ocean were taken aboard ships, but the type and size of ship could affect the measurement because size "determines the height at which observations were taken." Air temperature can change rapidly with height above the ocean. The authors therefore applied a correction for ship size in their data. Once Jones, Wigley, and Wright had made several of these kinds of corrections, they analyzed their data using a spatial averaging technique that placed measurements within grid cells on the Earth's surface in order to account for the fact that there were many more measurements taken on land than over the oceans.

Developing this grid required many decisions based on their experience and judgment, such as how large each grid cell needed to be and how to distribute the cells over the Earth. They then calculated the mean temperature within each grid cell, and combined all of these means to calculate a global average air temperature for each year. Statistical techniques such as averaging are commonly used in the research process and can help identify trends and relationships within and between datasets (see our Statistics in Science module). Once these spatially averaged global mean temperatures were calculated, the authors compared the means over time from 1861 to 1984.

A common method for analyzing data that occur in a series, such as temperature measurements over time, is to look at anomalies, or differences from a pre-defined reference value. In this case, the authors compared their temperature values to the mean of the years 1970-1979 (see Figure 3). This reference mean is subtracted from each annual mean to produce the jagged lines in Figure 3, which display positive or negative anomalies (values greater or less than zero). Though this may seem to be a circular or complex way to display these data, it is useful because the goal is to show change in mean temperatures rather than absolute values.

Figure 3: The black line shows global temperature anomalies, or differences between averaged yearly temperature measurements and the reference value for the entire globe. The smooth, red line is a filtered 10-year average. (Based on Figure 5 in Jones et al., 1986).

Putting data into a visual format can facilitate additional analysis (see our Using Graphs and Visual Data module). Figure 3 shows a lot of variability in the data: There are a number of spikes and dips in global temperature throughout the period examined. It can be challenging to see trends in data that have so much variability our eyes are drawn to the extreme values in the jagged lines like the large spike in temperature around 1876 or the significant dip around 1918. However, these extremes do not necessarily reflect long-term trends in the data.

In order to more clearly see long-term patterns and trends, Jones and his co-authors used another processing technique and applied a filter to the data by calculating a 10-year running average to smooth the data. The smooth lines in the graph represent the filtered data. The smooth line follows the data closely, but it does not reach the extreme values.

Data processing and analysis are sometimes misinterpreted as manipulating data to achieve the desired results, but in reality, the goal of these methods is to make the data clearer, not to change it fundamentally. As described above, in addition to reporting data, scientists report the data processing and analysis methods they use when they publish their work (see our Understanding Scientific Journals and Articles module), allowing their peers the opportunity to assess both the raw data and the techniques used to analyze them.

Concluding remarks

Surveys of forensic practitioners regarding aspects of training, proficiency testing, procedures, methods, policies, contamination prevention, data collection and communication relating to forensic trace DNA have highlighted the need for improvements in these areas [135, 136]. A number of recent reports have recommended the need for substantially greater investment into forensic services related research and development [82, 219]. This review identifies how far we have come in the use of trace DNA in order to assist forensic investigations in recent years, but it also identifies several opportunities for improvement in most facets of trace DNA work. A deeper consideration of workflow processes and priorities may yield alternative protocols that allow the use of a greater portion of the available DNA, with greater sensitivity, thus increasing the chance of generating fuller and easier to interpret profiles. Further research will improve the utilisation and benefits of collecting and typing trace DNA in forensic investigations.

Watch the video: Differentialligninger - hvordan løser man en differentialligning? (June 2022).


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