Mathematical models of pancreatic islet size distributions

Abstract

The islets of Langerhans, ranging in size from clusters of a few cells to several thousand cells, are scattered near large blood vessels. While the β-cell mass in mammals is proportional to body weight, the size ranges of islets are similar between species with different body sizes, possibly reflecting an optimal functional size. The large range of islet sizes suggests a stochastic developmental process. It is not fully understood how islets develop to reach such size distributions, and how their sizes change under certain physiological and pathological conditions such as development, pregnancy, aging, obesity, and diabetes. The lack of a high-resolution in vivo imaging technique for pancreatic islets implies that the only data available to elucidate the dynamics of islet development are cross-sectional quantifications of islet size distributions. In this review, we infer biological processes affecting islet morphology in the large by examining changes of islet size distributions. Neonatal islet formation and growth is shown as a particular example of developing a mathematical model of islet size distribution. Application of this modeling to elucidate islet changes under other conditions is also discussed.

Introduction

Although the islets of Langerhans occupy only 1–2% volume of the pancreas, they play a critical role in the homeostasis of blood glucose. Islets have three major endocrine cell types. α- and β-cells secrete glucagon and insulin for counteracting low and high glucose levels, respectively. δ-cells secrete somatostatin which inhibits both glucagon and insulin secretion.¹ Islets range in size from clusters of a few cells to several thousand cells. Rodent islets have a characteristic structure where β-cells are located in the islet core, while α- and δ-cells are in the periphery. In humans, on the other hand, non-β-cells are also distributed within the islet core.^2-5 Beyond such observations, we do not understand the design principles of islet architecture, i.e., the functional importance of islet size and cell arrangement in islets. In this review, we focus on the dynamic changes of islet size based on islet size distribution.

Understanding how islets form and grow in embryonic/neonatal development and change in pregnancy, aging, obesity or type 1 and type 2 diabetes is important because islet number and size are directly related to the change of β-cell mass. However, it is technically intractable to examine the in vivo formation of new islets and the proliferation potential of islet cells. Since real-time monitoring of changes in islets is currently not possible, most studies have relied on cross-sectional data from different animals at different conditions. In such studies, it is necessary to extract significant changes between the different conditions that may be masked by large variations between individual animals. Changes in islet number and size can be statistically represented by changes in islet size distribution. In principle, therefore, we can examine biological processes in islets in various conditions by mathematically analyzing changes in islet size distributions.

A prerequisite for such analyses is the precise measurement of islet sizes. Three ways are to measure size of islets (1) on an appropriately stained pancreatic section;⁶ (2) after isolation from a pancreas;⁷ and (3) in an intact pancreas.^8-10 The first method has been used most frequently. Although this immunohistochemical method has the advantages, allowing high resolution imaging of islets on 2-D pancreatic sections and multiple stains for different molecules, it has the fundamental limitation of revealing only randomly placed cross-sections through islets. To quantitate characteristics of 3-D islets, the second method uses isolated islets from the exocrine tissue. However, some small islets and endocrine cell clusters are likely to be lost during the isolation process. The last method is ideal for measuring size of entire islets without any modification. One difficulty is to distinguish islets from the exocrine pancreas. Dithizone and insulin antibody have been used to stain Zn²⁺ and insulin in β-cells.^8,10,11 Another novel tool is to use a transgenic mouse line in which β-cells are specifically labeled with green fluorescent protein (GFP) under the control of mouse insulin I promoter (MIP).^12,13 An excised pancreas from the MIP-GFP mouse is placed between a glass slide and a coverslip to recapture the endogenous anatomy of the pancreas.^9,14 Then, the entire distribution of islets is captured as an integrated 2-D bird’s eye view. Note that once a pancreas loses its structural retention from the tight connection to the surrounding gut, duodenum, colon and spleen, it naturally shrinks into a lump of soft tissue. Alanentalo et al. utilize optical projection tomography to measure islet size in an intact pancreas that is placed in an agarose gel to preserve its three-dimensional structure.¹⁰ Combined with these novel techniques, recent image analysis tools allow precise and automated measurement of islet size, morphology and location in a pancreas.^9,14 An exciting development in the field is experimental animal platforms for in vivo imaging of islets, although sufficiently high-resolution imaging has yet to be achieved. These approaches are comprehensively described in a recent review.¹⁵

In this review, we describe how we apply mathematical modeling to infer biological processes based on changes of islet size distributions to overcome experimental limitations and extract dynamic information. We have applied similar mathematical methods to analyze temporal changes of size distributions of fat cells under weight gain and loss conditions.^16,17

Islet Size Distribution

For the analysis of islet size, the islet diameter or area has been typically used as an estimate of islet size. The data are then plotted as a histogram of islet sizes. Figure 1 shows islet size distributions at various conditions. In particular, Figure 1A shows the size distribution of islets in a pancreas at day 1 after birth. Islet size varies from a few cells to large clusters of several thousand cells. Small clusters of endocrine cells are observed throughout the life regardless of species. Although the reason for the numerous small islets is unknown, blood-flow analysis suggests that they function to provide hormone rich portal blood to the acinar pancreas.¹⁸ Note that more than 50% of total β-cell area in a pancreas comes from islets with diameters in the largest 2% only (Fig. 1B). Furthermore, the marked size difference between the smallest and the largest islets is difficult to explain solely by the stochastic process of cell replication. For example, suppose a single cell seeds for an islet at embryonic day 10.¹⁹ Considering the gestational period of ~20 d for mice, this mother cell can replicate during the remaining 10 d. Assuming that every daughter cell replicates each day, the islet, starting from the mother cell, will have ~1,000 (2¹⁰) cells when the mouse is born. If the mother cell replicates only once or twice during the 10 d, it remains as a cluster of four or fewer cells at birth. It is unlikely that such a large heterogeneity in replication is the cause of the marked size difference between the smallest and the largest islets. Aggregation of cells or cell clusters is more plausible to explain the existence of very large islets. Indeed, migration and clustering of β-cells have been recorded during in vitro growth of the mouse embryonic pancreatic epithelium,²⁰ leading to cord-like branching structures. In addition, the polyclonal origin of islets in chimeric mice suggests that islets do not originate from a single seed.^21,22

Figure 1. Islet size distributions. (A) Absolute frequencies of effective diameters of islets in mouse pancreata at postnatal day 1 (P1, n = 8). Note that effective diameter of an elongated islet is defined so that the effective circle has the same area of the original islet area. Using the same data, re-plotted were (B) absolute frequencies of logarithmic islet area scaled by single-cell area, 170 μm² (red) and total area covered by all islets in each size bin (gray). Islet size distributions of mice (C) at P1 (red, n = 8), P14 (black, n = 6) and P21 (blue, n = 3); and (D) at week 24 (black, n = 14) and week 32 (blue, n = 10) (27). (E) Islet size distributions of 3-mo-old female mice (black, n = 6) and pregnant mice (blue, n = 5) (11). (F) Islet size distributions of non-obese diabetic (NOD) mice at 3 weeks (black, n = 6) and 12 weeks (blue, n = 6) (28). Note that islet volume is scaled by single-cell volume, 1,700 μm³. For comparison between different islet size scales, effective diameters, corresponding to each size bin, were given in parentheses. Error bars represent standard deviations.

Newborn mice have various sizes of islets. The distribution has an asymmetric shape skewed to smaller size, which differs from the Gaussian distribution that is frequently observed in nature (Fig. 1A). This skewed distribution of islet size has been fitted with various functional forms such as lognormal,^7,23 Weibull,^7,24 and inverse Gaussian functions.⁸ It is important to know which functions better fit the real size distributions of islets because the functional form is tightly related to the processes governing islet growth.^7,25 Considering the frequency of small islets and the rarity of large islets with equal significance, we propose to use logarithmic size bins to describe the islet size distribution (Fig. 1B). In the logarithmic size scale, the skewed lognormal distribution of islet diameter then becomes a Gaussian distribution, missing left tail. In general, the shapes of size histograms are considerably different for different indices of islet size, because islet diameter, area, and volume are scaled differently as d, d², and d³. The histogram of islet volumes must have the heaviest right tail. However, their shapes become the same in the logarithmic size scale, because their size scales of the diameter, area, and volume all scale (up to an irrelevant constant) as log(d) (Fig. 1). The islet size distribution has different shapes at different developmental stages (Fig. 1C and D), during pregnancy (Fig. 1E), and under diabetic conditions (Fig. 1F). In the following section, we describe how to extract dynamic information from changes of the static size distributions at different time points or conditions.

Changes in Islet Size Distribution

Any biological processes affecting islet sizes should lead to corresponding changes of islet size distributions. Possible biological processes, affecting islet size, are schematically described in Figure 2. Islet birth and growth occur mostly during embryonic and postnatal development.²⁶ During embryonic stage, active proliferation of endocrine cells may result in aggregation (fusion) of neighboring cells.²⁰ Interestingly, we have reported that large islets divide (fission) after birth, most actively at the weaning period.²⁷ In type 1 diabetes, islets are destructed by lymphocytes.²⁸ Damaged islets in diabetes may be eliminated or shrunken in size by cell death within islets. Figure 3 illustrates several examples of changes of islet size distribution by islet birth, fusion, (a)symmetric fission, size-(in)dependent growth/shrinkage. If we know about these processes, it is straightforward to predict changes of islet size distribution. However, we frequently meet the opposite condition that we know the changes of islet size distribution with lack of the precise information regarding the biological processes. Our goal, therefore, is to solve the inverse problem of estimating these processes based on the observed changes of islet size distribution from initial to final time points. Here we provide a schematic conceptual description of the method (Fig. 4).

Figure 2. Possible biological processes affecting islet size. Examples are recruitment of new islets at the minimal size s₀ with birth rate b; elimination of islets (size s₀) with death rate d₀; growth of islets (size s₀ and s₁) with growth rates and ; shrinkage of islets (size s₁ and s₂) with shrinkage rates and ; fusion of two islets (size s₀ and s₁) into one islet (size s₂) with fusion rate ; fission of one islet (size s₂) into two islets (size s₀ and s₁) with fission rates and , respectively.

Figure 3. Changes of islet size distribution. Black histograms represent an initial size distribution. Blue histograms represent final size distributions after given biological processes. Considered are several biological processes: islet birth, fusion, (a)symmetric fission, and size-(in)dependent growth and shrinkage.

Figure 4. Flowchart of the mathematical modeling. Initial size distribution and randomly-selected model parameters are put into the mathematical model that describes changes of size distribution. Then, the model predicts a final size distribution based on the given model parameter. The difference between predicted and measured final size distribution is quantified as a cost, which is scaled by measurement uncertainty. If the cost is less than a previous cost, we update the optimal model parameters with the new randomly-selected parameters. We iterate this process to find the most optimal parameters. More sophisticated explanation of the modeling is given in Appendix.

By incorporating one or more of the basic developmental processes mentioned above, we can construct a mathematical model that describes changes of islet size distribution. However, we do not know the values of various rates in the model such as the islet birth and growth rate. Our question is what parameter values are the best for describing measured changes of islet size distributions, and what their uncertainties are for the given data. First, we randomly assign parameter values (e.g., birth and growth rates) within physiological ranges. Second, we use a measured initial size distribution as the initial condition for evolving in the time course using the model with the randomly selected parameter values. The model predicts final size distributions as a function of the parameter values. Third, we compare the predicted size distribution with an observed final size distribution. The prediction with the random parameter values may not coincide with the experimental observation. The mismatch between predicted and measured distribution is quantified as a cost of fit. Note that we consider measurement uncertainties in the calculation of the cost. As the measurement uncertainties become larger, the mismatch between prediction and measurement is considered as less significant. Therefore, a lower cost of fit means that the given parameter values better explain the changes of islet size distribution. Finally, we iterate this procedure a number of times (usually millions of times) to obtain the maximum-likelihood parameter values that give the lowest cost. To search optimal parameter values efficiently, we use parallel tempering Monte Carlo simulation (see Appendix for more detailed explanation). In addition to the maximum-likelihood parameter values, the simulation allows us to estimate ranges of likelihood parameter values and compare between different models (e.g, size-dependent growth vs. size-independent growth).

Neonatal Islet Development: An Example

We have shown that endocrine cells proliferate contiguously, forming branched cord-like structures in embryos and neonates.¹³ Our follow-up study revealed long stretches of interconnected islets along large blood vessels in the neonatal pancreas, and we proposed a new model for islet formation that accounts for the morphological transformation from embryos to adults.²⁶ We have then developed a mathematical model of the dynamic process of islet development that islets are formed by random fission of large interconnected islet structures, which occurs most actively around the time of weaning postnatal day 21 (P21).²⁷Figure 1C shows islet size distributions at P14 and P21. Here we describe the change of islet size distribution in terms of islet birth, growth, and fission. We specifically compared three models: Model 1 (no fission) excludes the occurrence of islet fission; Model 2 (symmetric fission) considers that an islet divides symmetrically; and Model 3 (random fission) considers that an islet can divide into two islets of any sizes. The fission rate, , determines the way that an islet of size s_j splits into an islet of size s_i. The specific forms of for the symmetric and random fission are described in the appendix. Based on the three models, we have estimated islet birth, size-dependent growth and fission rates for each model (Fig. 5). Regardless of the models, the growth rate consistently showed that smaller islets grow faster than larger islets. Furthermore, the two fission models demonstrated that larger islets are more susceptible to fission, although the specific manner of fission is totally different between symmetric and random fission. Based on the predicted islet size distribution at P21, fission models generally fit the measured islet size distribution better than the model excluding islet fission. However, the random fission model fits the result better than the symmetric fission model. This has been more rigorously checked with a Bayesian model comparison. The improved fit with the random fission process can also be justified with the Akaike information criterion that has a penalty for increasing number of free parameters in models. Seymour et al. have proposed islet fission as a symmetric division of a large islet into two islets.²⁹ In contrast, we proposed that random islet fission is a process for forming mature spherical islets from long stretches of interconnected islets. Finally, the preferential growth of small islets and fission of large interconnected islet-like structures are potential mechanisms that maintain islet size within a certain range. We have shown that different species have a similar islet size range despite differences in body size.³⁰

Figure 5. Models for the postnatal development of islets. Based on the change of islet size distribution from postnatal day 14 to 21 (see Fig. 1C), islet birth, size-dependent growth and fission rates were estimated. Considered are three models: (1) Model 1 excludes the occurrence of islet fission; (2) Model 2 considers symmetric fission of islets; and (3) Model 3 considers random fission of islets in which an islet can divide into any sizes. Three models assumed the same size dependence of islet growth with an exponential function. Models 2 and 3 assumed the same size dependence of islet fission with a sigmoidal function. Here, 1, 2 and 3 free parameters were used to describe the islet birth, size-dependent growth and fission rates, respectively. Total numbers of parameters, therefore, were 3 (Model 1) and 6 (Models 2 and 3). Error bars were estimated from the standard deviation. The predicted size distributions are the most-likelihood fits of each model. Log-likelihood of each model was obtained by the relative Bayesian model probabilities, log[P(D|Model)/P(D|Model3)], which were calculated by parallel tempering Monte-Carlo(MC) simulation with 10⁶ MC steps.

Islet Size Distributions in Various Conditions

Pregnancy

Pregnancy is a unique physiological condition with tightly controlled rapid expansion and reduction of β-cell mass in a short period of time in the adulthood. It has been reported that β-cell mass increases 2- to 5-fold during pregnancy in rodents, which is accompanied by increasing cell replication and decreasing cell apoptosis.^13,30,31 However, a recent human study shows that islet neogenesis is the primary mechanism for β-cell expansion rather than the β-cell replication.³¹ Therefore, it is interesting to examine changes in the islet size distribution during pregnancy in different species, and see how β-cell mass expands in terms of recruitment of new islets and growth of existing islets, and the size dependence of this expansion. We applied our method to analyze changes of islet size distribution during pregnancy in mice,¹¹ and found that small islets expanded faster than larger islets (Fig. 6). Note that the size dependence in the islet growth is consistent with postnatal islet development (Fig. 5). Here the recovery process at postpartum is also interesting to see the shrinkage of islets that may also depend on islet size.

Figure 6. Models for islet growth during pregnancy. Based on the change of islet size distribution (see Fig. 1E), islet growth rate was estimated. Considered are two models: Model 1, constant growth independent on islet size; and Model 2, size-dependent growth assuming an exponential function. The total numbers of free parameters were 1 (Model 1) and 2 (Model 2). Error bars were estimated from the standard deviation. Note that they are tiny in the growth rates. The predicted size distributions are the most-likelihood fits of each model. Log-likelihood of each model was obtained by the relative Bayesian model probabilities, log[P(D|Model)/P(D|Model2)], which were calculated by parallel tempering Monte-Carlo(MC) simulation with 10⁶ MC steps.

Aging

Islet recruitment and growth are limited in adulthood. In particular, it has been reported that the main source of new β-cells during the late period is not neogenesis from stem/progenitor cells, but β-cell replication.^32,33 In addition to the rarity or lack of β-cell neogenesis, existing β cells replicate slowly (1 in ~1,400 β-cells per day) in adult mice.³⁴ Primate and human β-cells also replicate very slowly in adulthood.³⁵ The lack of β-cell neogenesis and slow replication lead to no recruitment of new islets and limited growth of islets, respectively. Consistent with these findings, islet size distribution does not change in 4–8 mo-old mice in our study.²⁷

Type 1 diabetes

Type 1 diabetes (T1D) is an autoimmune disease characterized by the destruction of insulin-producing β cells.³⁶ Non-obese diabetic (NOD) mice have been largely used as an animal model for T1D.³⁷ Recently, Alanentalo et al. have measured islet size distributions during the T1D progression in the NOD mice by optical projection tomography, and reported that there is a preferential loss of smaller islets, and a possible regeneration process in larger islets to compensate for the β-cell destruction.²⁸

Type 2 diabetes

Unlike T1D, type 2 diabetes (T2D) has an unclear pathogenesis and a slow disease progression, which makes it difficult to diagnose and efficiently treat patients with T2D. Obesity is associated with the development of T2D.³⁸ However, not all obese individuals are diabetic.³⁹ This suggests that T2D is a heterogeneous disorder. In human autopsy studies, it has been reported that β-cell mass decreased in subjects with T2D.^40,41 In particular, we have measured islet size distributions in subjects with T2D, and found that larger islets were preferentially lost in T2D,⁴² which is consistent with a previous report.⁴¹

Obesity

Obesity increases β-cell mass in rodents^43,44 and humans.^41,45 Bock et al. have reported that β-cell mass increases by islet expansion, but not by recruitment of new islets in ob/ob mice.⁶ Here the islet-size dependence of the expansion in obesity is interesting to see if it is consistent with the islet growth during postnatal development (Fig. 5) and pregnancy (Fig. 6). Both conditions have shown preferential growth of smaller islets; in other words, cells in smaller islets replicate more frequently than the ones in larger islets.

Discussion

Mathematical modeling can extract dynamic information from changes of static size distributions. It is difficult to observe dynamic phenomena, such as growth, fusion, and fission, experimentally without real-time monitoring. Recent novel imaging methods provide precise size measurements at specific times, providing a set of size distributions for inference. This review introduced how to analyze the changes of islet size distribution and understand underlying biological processes such as islet birth, growth, and fission. Unlike a simple analysis of mean size, analysis of the entire size distribution can provide size-dependent properties of biological processes. For instances, smaller islets have a higher probability to expand, while larger islets have a higher probability to undergo fission during the postnatal period.²⁷ In addition to these qualitative features of islet development, mathematical modeling can provide quantitative values of the biological processes. For example, the islet birth and fission rates allow us to estimate that 180 new islets resulted from birth and 40 new islets resulted from fission during the postnatal period between P14 and P21. Note that it is not an easy task to distinguish these distinct contributions to the appearance of 220 new islets with experiments. This method can be widely applied to analyze any size distributions. In particular, we have used it in the analysis of size distribution of fat cells, and found that larger fat cells are less efficient for storing and releasing fat.^16,17

For comprehensive modeling, two main difficulties exist. First, absolute (not relative) frequencies of islet sizes are required to precisely examine underlying biological processes including recruitment of new islets. However, counting every islet in a pancreas is impractical, especially in experiments with large animals such as humans. Alternatively, when we know islet density in a pancreas and weight of the pancreas, we can estimate total islet number, though adjustments must be made for the heterogeneity of the pancreas. Then, we can reconstruct absolute islet size distribution by multiplying the total islet number with the relative size distribution measured in a part of the pancreas. If no information except for the relative size distribution is available as usual in human pancreatic sections, the total islet number can be set as a free parameter and estimated to fit the given data best.

Second, individual variation, which is inevitable given the cross-sectional nature of islet size distribution, should be considered. In particular, a large individual variation of islet size distribution can mask changes of islet size distribution. Therefore, the modeling can only explain changes of averaging islet size distribution of large samples, although the individual heterogeneity is also an interesting subject. Real-time monitoring of islet development in a live individual animal may be the ultimate way to overcome the limitations of cross-sectional studies. Real-time imaging has been used to examine the growth of the embryonic pancreatic epithelium in culture.²⁰ However, in vivo imaging is still not feasible in large animals, unlike transparent zebra fish.⁴⁶

In this review, we have focused on islet sizes assuming islets have homogeneous cell types, presumably β-cells. However, real islets have different cell types, such as α-, β- and δ-cells, and different architectures in different species.³⁰ Therefore, islet development is a more complicated process including changes of cellular composition and arrangement in addition to the size. Islet architectures may be optimized for islet function. Coupling between homologous β-cells and communication between heterologous α-, β-, and δ-cells have been reported to have a potential role for secreting glucose-regulating hormones efficiently.^47,48 Recent advances in technologies for imaging and electrophysiology provides us many clues to understand how this micro-organ plays the central role of glucose homeostasis.^49,50 Mathematical modeling can contribute by integrating the morphological and physiological aspects of pancreatic islets.

Appendix: Mathematical Modeling

Number and size of pancreatic islets are changed by islet birth/death, growth/shrinkage and fusion/fission processes in physiological and pathological conditions. The change can be generally described by histograms of islet sizes. If the underlying biology is known, the evolution of the islet size distribution is easily computed. More frequently, the problem of interest is the inverse problem, to infer the biological processes from given intermediate-time islet size distributions. From a mathematical perspective, descriptions of biological processes can be made arbitrarily mathematically complex. This is inappropriate for data-driven modeling, as predictive models must balance goodness-of-fit against model complexity. Available data on islet size distributions does not yet justify any model features beyond simple protean growth and development processes. Here we introduce a general framework to solve such inverse problems. First, we start with a difference equation that describes changes of particle (islet) size distribution, n_i, the frequency of islets of size s_i (i = 0, 1, 2, ...):

(1)

This equation includes six possible processes in sequence:

(1) New particles appear with the birth rate, b, only at the minimal size s₀ which is described by the Kronecker-Delta function, δ_i,0 = 1 only if i = 0, otherwise 0.

(2) Particles disappear with the death rate, d_i, depending on size s_i. Rates are defined as a probability that an event (death) occurs for a single particle per unit time. Therefore, the total particle disappearance at a given size is calculated by multiplying the total particle number at the size with the death rate.

(3) Particles grow with the size-dependent growth rate, . The particle frequency at size s_i increases by the growth of smaller particles at size s_i-1, and decreases by the growth of the particles at the same size s_i.

(4) Particles shrink with the size-dependent shrinkage rate, . Similarly to the growth process, the particle frequency at size s_i increases by the shrinkage of larger particles at size s_i+1, and decreases by the shrinkage of the particles at the same size s_i.

(5) Two particles at sizes s_j and s_k fusion and make one bigger particle at size s_i ( = s_j +s_k) with the fusion rate, . For the given size s_i, while the fusion of smaller particles contributes to increase the particle frequency at the size s_i, the fusion of the particles at the size s_i with other islets decreases the frequency.

(6) A particle at size s_j fission into two particles at sizes s_i and s_k (> s_i) with the fission rate, . Again for a given size s_i, while the fission of larger islets contributes to increase the particle frequency at the size s_i, the fission of the particles at the size s_i decreases the frequency.

All the simplest general mechanisms for changing particle size distribution are included in Equation 1. Indeed, all six processes can occur in islet development and pathological conditions for pancreatic islets.

Our goal is to estimate the biological processes quantitatively based on the observed changes of islet size distribution. Here the processes become model parameters, e.g., birth rate, growth rate, etc. First, we randomly choose parameter values within reasonable rages, and then evolve Equation1 starting from a given initial size distribution with the selected parameter values. Finally, we get a predicted final distribution, n_i. It can be compared with a measured final distribution, , including measurement uncertainty, . The mismatch between the predicted and measured distribution can be quantified as a cost:

, (2)

where = (n₀, n₁, ...) is abbreviated for simplicity.

Here minimizing the cost (energy in statistical physics) is equivalent to improving the fit to the experimental result with maximum-likelihood estimates of model parameters. For a mean and variance , the probability of a configuration n_i is given by the maximum entropy principle:⁵¹

. (3)

Then, we used the Metropolis algorithm, a Markov Chain Monte Carlo (MCMC) method, to minimize the cost with the updating probability that guarantees the above equilibrium probability distribution p:

(4)

with .

Therefore, during the MCMC simulation, the configuration , depending on parameter , changes and automatically satisfies the equilibrium configuration. Here the parameter vector is defined as . Actually we used a functional form instead of treating all the size-dependent rates independently. For example, the growth rate has been described as an exponential form . Therefore, are actual parameters for the growth instead of g_i for every size s_i (i = 0, 1, ...). For describing islet fission, we used where q_ij describes the way that an islet of size s_i fission into an islet of size s_i, and the f_i controls the fission rate of the islet of the size s_j. In particular, symmetric fission is described by q_ij = 2δ_i,j-1 which allows only s_j fission into two s_j-1 islets. Note that logarithmic size bin is used for this description; when area A fission into two area A/2 particles, the logarithmic size becomes s_i = log₂A and s_i = log₂(A/2) = s_i -1 = s_i-1. On the other hand, random fission is described by q_ij = 2·2ⁱ/(2^j-1). This corresponds to a process in which an islet of size s_j can divide into any size s_i. However, technically in the logarithmic size bin, upper size bin covers larger size range with the scale 2ⁱ for the size s_i bin. The factor (2^j-1) in denominator is just a normalization factor from . Both cases assume binary fission with . We have used a sigmoidal function for f_j to describe size-dependent fission rate²⁷: . Now, our actual parameters for islet fission become instead of f_j for every size s_j (j = 0, 1, ...). The configuration is ultimately determined by the parameter. Therefore, we are actually wandering in the parameter space, not in the configuration space during the MCMC simulation. This allows us to estimate (1) the optimal parameters corresponding to the equilibrium configuration, and (2) parameter uncertainties based on the fluctuation near the equilibrium:

(5)

where N is total MC steps.

Practically, we used parallel tempering MCMC⁵² where multiple levels of fluctuations with different temperatures are used to take the advantage of global searching with high temperature, and of fine searching with low temperature. Therefore, we run multiple MC chains with different temperatures, and exchange configurations (parameters) between two chains with a given mixture rates. Here we mix the configurations at different temperature simulations with the updating probability:

. (6)

Therefore, the parameter exchange between two chains is only allowed based on the difference of the global cost between (after the exchange) and (before the exchange).

Another big advantage using parallel tempering MCMC is to compare different models. A partition function of the cost is defined as:

. (7)

This partition function exactly corresponds to the Bayesian model probability of how likely a given model is, given observed data:

with Θ = 1. (8)

The following relation,

, (9)

gives

. (10)

Therefore, the Bayesian model probability becomes

with Θ_i = i/m and ΔΘ = 1/m. (11)

This quantity roughly represents how well the given model fits the data over the entire parameter space. For example, usually using more parameters can fit better the data, but there is a complexity cost associated with additional parameters. This Bayesian model probability permits a balance between goodness-of-fit and model complexity, avoiding over-fitting the available data.