Mathematical Modeling of the Gut–Bone Axis and Implications of Butyrate Treatment on Osteoimmunology

Abstract

Butyrate, a short-chain fatty acid produced by the gut microbiota, has pivotal roles in the regulation of the immune system. Recent studies have revealed that butyrate increases the differentiation of peripheral regulatory T cells in the gut–bone axis and promotes osteoblasts’ bone forming activity. However, the mechanism of the therapeutic benefit of butyrate in bone remodeling remains incompletely understood. Here, we develop a multicompartment mathematical model to quantitatively predict the contribution of butyrate on the expansion of regulatory T cells in the gut, blood, and bone compartments. We investigate the interplay between regulatory T cell-derived TGF-β and CD8+ T cell-derived Wnt-10b with changes in gut butyrate concentration. In addition, we connect our model to a detailed model of bone metabolism to study the impacts of butyrate and Wnt-10b on trabecular bone volume. Our results indicate both direct and indirect immune-mediated impacts of butyrate on bone metabolism.

Graphical Abstract

INTRODUCTION

Gut microbiota plays a major role in regulating nutrient availability, metabolism, and immune function by facilitating the digestion, absorption, and secretion of various secondary metabolites. Recent evidence has revealed that the gut microbiota regulates bone homeostasis in health and disease.^1–5 The manipulation of intestinal bacteria through the use of antibiotics, dietary interventions, and germ-free mouse models altered the bone density and microarchitecture.^6–10 These results give insight into potential therapeutic benefits as humans and mice share qualitatively similar gut microbiomes.¹¹ Fermentation of dietary fiber generates metabolites critical for biological activity. Short-chain fatty acids (SCFAs) have received the most attention among the metabolites for their potent regulatory effect locally and on distant organs.^12,13 SCFAs have pro- and anti-inflammatory activity that modulates local and systemic immune responses.¹⁴ Their ability to modulate histone deacetylases and activate G-protein-coupled receptors has important functions in metabolism, inflammation, and disease.¹⁵ The amounts of SCFAs produced in the gut can be modified generally through either increasing the prebiotic fiber content of the diet (i.e., more raw materials for the existing gut microbes to convert to SCFAs) or supplementing the diet with probiotics or other bacterial transfer approaches that alter the populations in the gut microbial community to favor enhanced production of SCFAs (i.e., more total microbes and/ or more microbes that enhance the fermentation rate of fibers to convert the same amount of raw materials to SCFAs with higher throughput). Additionally, SCFAs can be directly supplemented.¹⁶

Recently, the SCFA butyrate has emerged as a regulator for bone resorption and bone formation.^17,18 The therapeutic efficacy of butyrate for immune-mediated inflammatory diseases, such as inflammatory bowel disease,¹⁹ diet-induced obesity, and insulin resistance,^20,21 and proteinuria and chronic kidney disease,^22,23 has been demonstrated in animal models. Butyrate also attenuates rheumatoid arthritis, controls the balance between T helper 17 cells (Th17) and regulatory T cells (Tregs), inhibits the differentiation of osteoclasts and bone resorption,¹⁶ and inhibits osteoclastogenesis in mouse models.²⁴ Studies have indicated that butyrate stimulates bone formation in the gut–bone axis^17,25 and is pivotal for parathyroid hormone-dependent bone formation.^26,27

Butyrate promotes the differentiation and proliferation of Tregs from naïve CD4+ T cells locally in the gut and systemically.^28–30 Tregs suppress the activity of other T cells and maintain immune tolerance. There are two complementary classes of Tregs: thymus derived natural Tregs (nTregs) and peripheral Tregs (pTregs).³¹ Butyrate increases the number of pTregs in the intestine, blood, and bone marrow. Tregs exert an immunomodulatory effect by producing and secreting immunosuppressive cytokines, such as TGF-β and IL-10.^32,33 Interactions of Tregs with bone marrow CD8+T cells regulate the production of Wnt-10b via TGF-β signaling to promote bone anabolism, that is, net formation of bone resulting from the bone remodeling process.^18,34 The mechanism of Wnt-10b expression and immunosuppression promotes the differentiation of osteoblasts resulting in an increase in bone formation³⁵ and suppresses bone resorption.^36,37

Since butyrate increases bone mass directly and indirectly by interacting with bone and immune cells, respectively, it has the potential to be used as a treatment to restore bone health. Age-related bone loss is common, and treatment is expensive. According to a report from the National Osteoporosis Foundation, 44 M Americans have poor bone mass, 10 M Americans have osteoporosis, and the total economic impact is around $44 B annually.³⁸ Osteoporosis is a condition with devastating consequences, and progressive bone loss often results in fractures of the wrist, hip, or spine.³⁹ Most of the cases remain untreated or poorly treated due to the cost and side effects of the available drugs. Dietary interventions and other therapeutics that can be used to increase SCFA production or concentrations are promising alternatives for the prevention and treatment of osteoporosis.

Existing mathematical models that investigate the dynamics of butyrate have been limited to a single compartment and have not provided insight into the therapeutic application to bone health. Neumann et al. developed a dynamical model to investigate the robust bistability of anti-inflammatory butyrate and pro-inflammatory bacterial lipopolysaccharides in gut epithelial cells.⁴⁰ The model of Korsten et al. showed that orally administered butyrate is capable of achieving pharmaceutical responses in the small intestine through immediate and sustained release.⁴¹ Here, we develop a mathematical model to investigate the interaction between butyrate and immune cells in the gut–bone axis and how the interaction of the gut and bone can contribute to changing trabecular bone volume. In addition to these immune-mediated indirect effects of butyrate, we also conduct an in silico experiment to determine possible mechanisms for direct impacts of butyrate on bone metabolism.

METHODOLOGY

Physiologically Based Pharmacokinetic Pharmacodynamic Model of the Gut–Bone Axis.

Here, we develop a multicompartment physiologically based pharmacokinetic pharmacodynamic model to track butyrate and quantify its effects on Tregs in three compartments: gut, blood, and bone (Figure 1). The model consists of ordinary differential equations (ODEs) for three species—butyrate, naïve CD4+ T cells, and Tregs—distributed across the three compartments and for two additional species—TGF-β and Wnt-10b—representing the pharmacodynamic effect in the bone compartment. We neglect any spatial aspects along the length of the intestinal tract. We assume that all three compartments can be approximated as well-mixed, constant volume systems. We consider an open system with the processes of formation, degradation, differentiation, cell death, degradation, and migration between compartments. We define mass balances on the chemicals butyrate, TGF-β, and Wnt-10b, and population balances for the naïve CD4+ T cells and Tregs. We define the symbols F_ij for source or formation terms, μ_i for death or degradation terms (assumed the same across all compartments for a given species), A_ijk for absorption terms, and δ_ijk for cell migration terms between compartments, where i represents the species, j represents the source compartment, and k represents the destination compartment. The numbers 1, 2, and 3 denote the gut, blood, and bone compartments, respectively.

Figure 1.

Diagram for the processes involved in the three-compartment physiologically based pharmacokinetic model for butyrate and its effects on bone remodeling via differentiation of naïve CD4+ T cells to Tregs and Wnt-10b signaling. The numbers 1, 2, and 3 denote the gut, blood, and bone compartments, respectively. Chemical species and cell populations tracked are illustrated as blue rounded rectangles. Yellow circles and ovals represent processes through which the species or cells interact, move, are dosed, degrade, or die. Solid arrows denote transfer of one species to another, and gray dotted lines denote interactions that influence processes without being produced or consumed.

Equations 1–3 describe the dynamics of the change in butyrate concentration from homeostasis in the gut (B₁), blood (B₂), and bone (B₃) compartments, respectively,

$$\frac{\text{d}{B}_1}{ \text{d}t}=F_{B1}-{\mu }_B{B}_1-A_{B12}{B}_1$$ (1)

$$\frac{\text{d}{B}_2}{ \text{d}t}=A_{B12}{B}_1-{\mu }_B{B}_2-A_{B23}{B}_2$$ (2)

$$\frac{\text{d}{B}_3}{ \text{d}t}=A_{B23}{B}_2-{\mu }_B{B}_3$$ (3)

where F_B1 is the zero-order source term for butyrate in the gut, μ_B is the degradation rate constant (lumped effect of degradation, loss to local tissue, and/or excretion via the urine or feces) of butyrate in all three compartments, A_B12 is the absorption of butyrate from the gut compartment to the blood compartment, and A_B23 is the absorption of butyrate from the blood compartment to the bone compartment. We consider deviation variables where the initial condition of homeostasis has values of B₁ = B₂ = B₃ = 0; that is, the absence of excess butyrate is considered to be homeostasis.

The populations of naïve CD4+ T cells in the gut (N₁), blood (N₂), and bone (N₃) compartments, respectively, are described by

$$\frac{\text{d}{N}_1}{ \text{d}t}=F_{N1}^{-}-{\mu }_N{N}_1-\gamma b^{-}{N}_1+F_{N1}^{+}-b^{+}{B}_1{N}_1$$ (4)

$$\frac{\text{d}{N}_2}{ \text{d}t}=F_{N2}^{-}-{\mu }_N{N}_2-\gamma b^{-}{N}_2+F_{N2}^{+}-b^{+}{B}_2{N}_2$$ (5)

$$\frac{\text{d}{N}_3}{ \text{d}t}=F_{N3}^{-}-{\mu }_N{N}_3-\gamma b^{-}{N}_3+F_{N3}^{+}-b^{+}{B}_3{N}_3$$ (6)

where $F_{Nj}^{-}$ is the formation rate of naïve CD4+ T cells in the absence of excess butyrate in each compartment j, μ_N is the death rate constant for naïve CD4+ T cells, the product γb⁻ is the rate constant for in vivo differentiation of naïve CD4+ T cells to Tregs in the absence of excess butyrate, F⁺_Nj is the formation rate of naïve CD4+ T cells in the presence of excess butyrate in each compartment j, and b⁺ is the rate constant for the differentiation of naïve CD4+ T cells to Tregs proportional to the change in butyrate concentration from homeostasis. We do not consider the circulation of CD4+ T cells between compartments. We parametrize the differentiation rate using in vitro data in which only butyrate is added to naïve CD4+ T cells in a culture medium well-suited for promoting differentiation (Supporting Information, Figure S1). The in vivo differentiation of Tregs also depends on the concentration of TGF-β and other physiological cytokines. Our model does not explicitly consider the effects of these cytokines on differentiation. Rather, the parameter γ is defined in the range of 0 to 1 to include the lumped effect of all the in vivo environmental components on modulating the suitability of conditions for differentiation, where a value of 0 is completely inhibited differentiation, and a value of 1 represents ideal conditions that are as well-suited as the in vitro culture medium.

The populations of Tregs in the gut (T₁), blood (T₂), and bone (T₃) compartments, respectively, are described by

$$\frac{\text{d}{T}_1}{ \text{d}t}=\gamma b^{-}{N}_1+b^{+}{B}_1{N}_1-{\mu }_T{T}_1-{\delta }_{T12}{T}_1$$ (7)

$$\frac{\text{d}{T}_2}{ \text{d}t}=\gamma b^{-}{N}_2+b^{+}{B}_2{N}_2+{\delta }_{T12}{T}_1-{\mu }_T{T}_2-{\delta }_{T23}{T}_2$$ (8)

$$\frac{\text{d}{T}_3}{ \text{d}t}=\gamma b^{-}{N}_3+b^{+}{B}_3{N}_3+{\delta }_{T23}{T}_2-{\mu }_T{T}_3$$ (9)

where μ_T is death rate constant for Tregs, δ_T12 is the migration of Tregs from gut to blood, and δ_T23 is the migration of Tregs from blood to bone. The ODEs for Tregs have production terms corresponding to the differentiation of naïve CD4+ T cells in each compartment in both the absence and the presence of excess butyrate.

Bone marrow Tregs release the cytokine TGF-β, which induces a complex signaling cascade. Here, we only consider the role of TGF-β in Wnt-10b upregulation. We denote the fold change of TGF-β as X_Tβ, defined as the ratio of the concentration of TGF-β at any time to the concentration of TGF-β at homeostasis, and X_Tβ = 1 at homeostasis. The rate of change of X_Tβ in the bone compartment is described by

$$\frac{\text{d}{X}_{T\beta }}{\text{d}t}={\mu }_{T\beta }\left(1-X_{T\beta }\right)+\frac{V_{T\beta }\left(\frac{T_3}{T_{3,0}}-1\right)}{k_{T\beta }-\left(\frac{T_3}{T_{3,0}}-1\right)}$$ (10)

where μ_Tβ is the degradation rate constant of TGF-β, and T_3,0 is the initial population of bone Tregs at homeostasis. We assume a mathematical form for the dynamics of the fold change of TGF-β due to change in bone Tregs similar to a Michaelis–Menten relationship, and V_Tβ and k_Tβ are the corresponding rate parameters.

CD8+ T cells release Wnt-10b when inflammation is suppressed in bone marrow via TGF-β signaling.¹⁸ Here, we consider the fold change of Wnt-10b as X_W, defined as the ratio of the concentration of Wnt-10b at any time to the concentration of Wnt-10b at homeostasis, and X_W = 1 at homeostasis. The rate of change of X_W due to the change in X_Tβ is

$$\frac{\text{d}{X}_W}{ \text{d}t}={\mu }_W\left(1-X_W\right)+{\rho }_W\left(X_{T\beta }-1\right)$$ (11)

where μ_W is the degradation rate constant of Wnt-10b, and ρ_W is the parameter for the correlation for changes in Wnt-10b mediated by changes in TGF-β.

Pharmacodynamic Impact of Butyrate on Bone Metabolism.

We connect our model to a detailed bone compartment model for bone metabolism that we previously developed⁴² to calculate the effects of intestinal butyrate concentration changes on trabecular bone volume mediated by the immune system throughout the three compartments. When we refer to trabecular bone volume, we mean the fraction of total bone volume that is composed of trabecular bone. Hence, the total volume for the bone compartment is constant, while the trabecular bone volume % is allowed to vary. The bone metabolism model⁴² estimates the change in trabecular bone volume for a constant Wnt-10b fold change for single or multiple bone remodeling cycles. Instead of constant Wnt-10b fold change, we modify the bone metabolism model⁴² to take as input the dynamic Wnt-10b fold change output from our model. The bone metabolism model⁴² assumes a 100 day bone remodeling cycle for humans, and we scale this down to the average 14 day remodeling cycle for mice as we parametrize our model using experimental data from mouse models. The updated form of the bone metabolism model equations are in the Supporting Information. Refer to the original bone metabolism model⁴² for the full set of parameter values and assumptions for that model. This connection between the bone metabolism model and our model determines immune-mediated indirect effects of butyrate. We conduct an in silico experiment to determine possible direct impacts of butyrate on bone metabolism. We define a series of mathematical cases modifying the bone metabolism model for hypothesis-generating in silico exploration of mechanisms by which butyrate may have direct effects in the bone compartment; these cases are defined in the Supporting Information.

Data Extraction.

To parametrize our model, the open-source software tool Plot Digitizer⁴³ is used to extract experimental data from graphs in the original sources cited in the text. In the following, we drop the word compartment when discussing the gut, blood, and/or bone compartments, and we refer to the gut as the intestine interchangeably.

Model Initialization and Input.

The model simulations are initialized at homeostatic conditions. We aim to simulate similar conditions as reported in experimental mice models. The details of model initialization at homeostasis, parameter estimation for butyrate and immune cells, and parameter estimation for species in the bone compartment are described in the Supporting Information. Note that we define F_B1 to be the zero-order source term of butyrate in the gut. This is used to represent (a) the impact of direct butyrate dosing on intestinal butyrate concentration, (b) the increased production of butyrate in the intestine via microbial fermentation of dietary fiber modulated by some prebiotic or probiotic treatment, or (c) the reduced production of butyrate in the intestine due to antibiotic treatment. For comparison to experimental studies, the actual dose of butyrate in the gut via various treatments is not known explicitly. Instead, we consider as model input specified new steady-state values of intestinal change in butyrate concentration due to an ongoing treatment B_1,ss and determine from eq 12 the corresponding step change in F_B1 to perturb the model away from homeostasis (Table 1):

$$F_{B1}={\mu }_B{B}_{1,\text{ss}}+A_{B12}{B}_{1,\text{ss}}$$ (12)

We calculate $F_{Nj}^{+}$ to replenish the population of naïve CD4+ T cells dependent on the butyrate dose. We rewrite eqs 4–6 considering steady-state conditions as follows:

$$0=F_{N1}^{-}-{\mu }_N{N}_{1,\text{ss}}-\gamma b^{-}{N}_{1,\text{ss}}+F_{N1}^{+}-b^{+}{B}_{1,\text{ss}}{N}_{1,\text{ss}}$$ (13)

$$0=F_{N2}^{-}-{\mu }_N{N}_{2,\text{ss}}-\gamma b^{-}{N}_{2,\text{ss}}+F_{N2}^{+}-b^{+}{B}_{2,\text{ss}}{N}_{2,\text{ss}}$$ (14)

$$0=F_{N3}^{-}-{\mu }_N{N}_{3,\text{ss}}-\gamma b^{-}{N}_{3,\text{ss}}+F_{N3}^{+}-b^{+}{B}_{3,\text{ss}}{N}_{3,\text{ss}}$$ (15)

We assume that at steady-state the naïve CD4+ T cells in each compartment j are restored to their initial homeostatic levels by way of changes in their source terms $F_{Nj}^{+}$ that depend on the butyrate concentration. Thus, N_j,ss = 1. Substituting the values for N_j,ss and other unknowns determined by eq S18 yields

$$F_{Nj}^{+}=b^{+}{B}_{j,\text{ss}}$$ (16)

for each compartment j. We rewrite eqs 2 and 3 considering steady state conditions to determine the corresponding values of B_2,ss and B_3,ss subject to a treatment value of B_1,ss:

$$B_{2,\text{ss}}=\frac{A_{B12}{B}_{1,\text{ss}}}{{\mu }_B+A_{B23}}$$ (17)

$$B_{3,\text{ss}}=\frac{A_{B23}{B}_{2,\text{ss}}}{{\mu }_B}$$ (18)

Then eqs 1–11 are solved with the other parameters listed in Table 2. We have provided our code in a repository at github.com/ashleefv/butyrateGutBoneAxis.⁴⁴

Table 1.

Initial Conditions for Model Variables, Where This Set of Values Is Referred to as the “Homeostasis” Condition

symbol	definition	initial condition
B 1	change of butyrate in gut	0 μM
B 2	change of butyrate in blood	0 μM
B 3	change of butyrate in bone	0 μM
N 1	Naïve CD4+ T cells in gut	1
N 2	Naïve CD4+ T cells in blood	1
N 3	Naïve CD4+ T cells in bone	1
T 1	Tregs in gut	0.086
T 2	Tregs in blood	0.15
T 3	Tregs in bone	0.72
X Tβ	TGF-β in bone	1
X w	Wnt-10b in bone	1

Table 2.

List of Parameters for the Model Defined by Equations 1–11

symbol	definition	value	unit	source
F B1	formation rate of butyrate in gut	dose dependent	μM day−1	calculated by eq 12
μ B	degradation rate constant of butyrate	166.3	day−1	refs 48 and 49
A B12	absorption rate constant of butyrate from gut to blood	5.36 × 102	day−1	calculated by eq S9
A B23	absorption rate constant of butyrate from blood to bone	1.66 × 102	day−1	calculated by eq S8
$F_N^{-}$	production rate of naïve CD4+ T cells in the absence of butyrate	4.05 × 10−2	day−1	calculated by eq S18
μ N	death rate constant of naïve CD4+ T cells	0.02	day−1	refs 50 and 51
γ	coefficient for similarity to Treg polarizing conditions	1.19 × 10−1	Unitless	eqs S3, S13–S14
b −	homeostasis differentiation rate constant of naïve CD4+ T cells	1.72 × 10−1	day−1	estimated from ref 28 data
$F_{Nj}^{+}$	production rate of naïve CD4+ T cells in the presence of excess butyrate	dose dependent	day−1	calculated by eq 16
b +	butyrate induced diff. rate constant of naïve CD4+ T cells	2.30 × 10−2	day−1 μM−1	calculated by eqs S19–S21
μ T	death rate constant of Tregs	0.064	day−1	ref 52
δ T312	migration rate of Tregs from gut to blood	1.73 × 10−1	day−1	calculated by eqs S3, S13–S14
δT23	mMigration rate of Tregs from blood to bone	1.73 × 10−1	day−1	calculated by eqs S3, S13–S14
V Tβ	rate parameter connecting Tregs to TGF-β fold change	6.00 × 10°	Unitless	estimated from refs 18 and 26 data
k Tβ	saturation parameter connecting Tregs to TGF-β fold change	4.89 × 10−1	Unitless	estimated from refs 18 and 26 data
μ Tβ	degradation rate constant of TGF-β	2	day−1	ref 51
ρ W	parameter connecting TGF-β to Wnt-10b fold change	1.64 × 10°	Unitless	estimated from ref 18 data
μ W	degradation rate constant of Wnt-10b	2	day−1	ref 51

Sensitivity Analysis.

We use local and global sensitivity analyses to quantify the input parameters that significantly affect the output variables. We vary b⁺, b⁻, γ, F_B1, A_B12, A_B23, δ_T12, and δ_T23 as input parameters to evaluate the sensitivity of the output variable bone Tregs (T₃). First, the local sensitivity is estimated by changing one input parameter by 10% at a time while the other parameters are held constant, and we compare the change in output variable T₃. Equation 19 shows the local sensitivity index for T₃ with respect to the varying model parameter (P_i), which is approximated by a small perturbation ΔP_i:

$$\frac{\delta T_3}{\delta P_i}=\underset{\Delta P_i\to +0}{\text{lim}}\frac{T_3\left(P_i+\Delta P_i,P_{n\ne i}\right)-T_3\left(P_i,P_{n\ne i}\right)}{\Delta P_i}$$ (19)

where T₃(P) is the model prediction of the bone Tregs at 28 days after treatment evaluated for parameter set P. We normalize the local sensitivity index to eliminate the effect of units:

$$S_i=\frac{\delta T_3}{\delta P_i}\frac{P_i}{T_3(P)}$$ (20)

We perform global sensitivity analysis to evaluate the effect of potential interactions of the input parameters in an output variable. We adopt the Sobol sensitivity analysis of the SALib package in Python to perform the global sensitivity analysis.⁴⁵ Input parameters are sampled using the Saltelli sampler.^46,47 We use the lower and upper bound of the parameter set as 0.1-fold and 10-fold of the baseline parameters, respectively. The first-order and total-order indices are estimated using the Sobol sensitivity analysis.

RESULTS AND DISCUSSION

Our model defined in eqs 1–11 was used to investigate the changes in butyrate, naïve CD4+ T cells, and Tregs in the gut, blood, and bone and fold changes of TGF-β and Wnt-10b in the bone compartment due to variations in intestinal butyrate concentration. In this section, we discuss the validation of the model for both treatment and pathological conditions and sensitivity analysis for model parameters. We also demonstrate the prediction of change in bone Tregs, TGF-β, and Wnt-10b with intestinal butyrate concentration that varies from homeostasis. We show that our predictions are in agreement with published literature. Finally, we analyze the impacts of butyrate and Wnt-10b on bone metabolism using our model simulations.

Parameter Estimation and Validation.

The values of the species at homeostasis are listed in Table 1. The estimated and calculated parameters are listed in Table 2.

We evaluated the biodistribution of butyrate and Tregs in the intestine, blood, and bone to compare to the experimental data reported in Tyagi et al.¹⁸ (Table S1). We initialized all of the simulations at the homeostasis conditions (Table 1). We simulated the model with an input of B_1,ss = 0.18 μM corresponding to the Lactobacillus rhamnosus GG (LGG) treatment stimulation of intestinal butyrate concentration. Figures 2 and 3 show the biodistribution of butyrate and Tregs, respectively, in the compartments. The data we used for parameter estimation are indicated in the figures by open red squares. If error bars are included, the data was determined from the source. If no bars are included, the data point was assumed to be the same as another reported quantity (e.g., B_2,ss = B_3,ss). While over the course of 28 days, the change in butyrate looks like a step change (Figure 2A–C), the model input only prescribes the steady-state values that result from perturbing away from homeostasis with a butyrate dose. The butyrate concentrations change dynamically within the first hour to approach the new steady state (Figure 2D–F). Tyagi et al.¹⁸ reported a 6% increase from the homeostasis value of the percentage of Tregs among CD4+ T cells in the bone marrow due to increase in intestinal butyrate concentration from LGG treatment (value T_3f,ss in Table S1). This data was not used in parameter estimation. Figure 3B shows a 5.2% increase of Tregs in the bone at 28 days, and there is good agreement between the shaded region and the error bars for the bone Tregs. This is considered as one model validation case.

Figure 2.

Change of butyrate concentration in (A) gut, (B) blood, and (C) bone in the first hour after simulated treatment with B_1,ss = 0.18 μM and in (D) gut, (E) blood, and (F) bone for 28 days of continuous simulated treatment. The shaded areas represent the range of solutions obtained by propagating the experimental variations (error bars) of intestinal butyrate concentration reported in Tyagi et al.¹⁸ through the simulations. Open red squares represent the experimental data or assumed data used in parameter estimation (see Table S1).

Figure 3.

Percentage of Tregs among CD4+ T cells in (A) blood and (B) bone for continuous simulated butyrate treatment with B_1,ss = 0.18 μM. The shaded areas represent the range of solutions obtained by propagating the experimental variations (error bars) of intestinal butyrate concentration reported in Tyagi et al.¹⁸ through the simulations. Open red squares represent the experimental data or assumed data used in parameter estimation, and the solid red circle represents data used for model validation (see Table S1).

We also evaluated the biodistribution of butyrate and Tregs in the intestine, blood, and bone to compare to the experimental data reported in Li et al.²⁶ (Table S2). We simulated the model with an input of B_1,ss = −0.11 μM corresponding to the antibiotic treatment suppression of intestinal butyrate concentration. We only changed the intestinal butyrate concentration and kept all other parameters the same as in the previous case. Figures 4 and 5 show the biodistribution of butyrate and Tregs, respectively, in the compartments. Again, the butyrate concentrations change dynamically, but the new steady state for butyrate is reached quickly. Li et al.²⁶ reported a 4.5% reduction from their homeostasis condition of the percentage of Tregs among CD4+ T cells in the bone marrow due to decrease in intestinal butyrate concentration from antibiotic treatment. We used that same reduction from our homeostasis condition to determine the value of T_3f,ss in Table S1. This data was not used in parameter estimation. Figure 5 shows a 3.7% reduction of Tregs in the bone at 28 days, and there is good agreement between the shaded region and the error bars for the bone Tregs. This is considered as another model validation case.

Figure 4.

Change of butyrate concentration in (A) gut, (B) blood, and (C) bone for 28 days of continuous simulated treatment with B_1,ss = −0.11 μM. The shaded areas represent the range of solutions obtained by propagating the experimental variations (error bars) of blood butyrate concentration reported in Li et al.²⁶ through the simulations. The open red square represents the experimental data used in parameter estimation (see Table S2).

Figure 5.

Percentage of Tregs among CD4+ T cells in bone for continuous simulated butyrate treatment with B_1,ss = −0.11 μM. The shaded area represents the range of solutions obtained by propagating the experimental variations (error bars) of blood butyrate concentration reported in Li et al.²⁶ through the simulations. The open red square represents the assumption of the same homeostasis calculated from experimental data of Tyagi et al.,¹⁸ and the solid red circle represents data used for model validation (see Table S2).

We performed a simultaneous iterative curve fitting using initial guesses (V_Tβ, k_Tβ) = (1, 1) to estimate the dynamics of TGF-β for both positive and negative changes of intestinal butyrate concentration. Here, we assumed that TGF-β has not reached a steady-state when the data are measured. The rationale for our assumption and details of the procedure for the simultaneous iterative curve fitting process to estimate parameters V_Tβ and k_Tβ are described in the Supporting Information. Figure 6 shows the dynamics of TGF-β for which we estimated the steady-state values (open circles) and fit the model (blue curves) with experimental data (solid circles, values $X_{T\beta ,m}^{+}$ and $X_{T\beta ,m}^{-}$ from Tables S1 and S2, respectively) for both positive (B_1,ss = 0.18 μM) and negative (B_1,ss = −0.11 μM) changes in the intestinal butyrate concentration. We assumed that t_m = 28 days for the Li et al.²⁶ data although they were actually obtained at a variety of times for antibiotic treatment (discussed in the Supporting Information). The lower curve in Figure 6 illustrates that there is little variation in this curve between 28 and 50 days, thus we considered the assumption of t_m = 28 days to be reasonable.

Figure 6.

TGF-β fold change with time due to intestinal butyrate concentration changes. The solid circles represent the experimental data points (values $X_{T\beta ,m}^{+}$ and $X_{T\beta ,m}^{-}$ from Tables S1 and S2, respectively) at positive (B_1,ss = 0.18 μM) and negative (B_1,ss = −0.11) changes in intestinal butyrate concentrations. The steady-state (S.S.) points represented by open circles were evaluated using simultaneous iterative curve fitting.

Local and Global Sensitivity Analyses.

The local and global sensitivity analyses were conducted to quantify the sensitivity of model output bone Tregs with input parameters. The analyses were performed for the parameters b⁺, b⁻, γ, F_B1, A_B12, A_B23, δ_T12, and δ_T23. Figure 7 shows the normalized local sensitivity indices, and Figure 8 shows the first-order and total-order global sensitivity indices for bone Tregs across the input parameter set. The results from both local and global sensitivity analyses showed that the output is more sensitive to parameters b⁻, b⁺, γ, and F_B1. These parameters were estimated or calculated from experimental results. We can perturb these parameters by changing experimental conditions to develop new treatments for pathological conditions. For example, environmental cytokines can be modified to differentiate more naïve CD4+ T cells to Tregs, which will change γ, or we can increase the probiotic or direct butyrate dose to change the amount of butyrate in the intestine, which will change F_B1 in our model. The sensitivity analyses also showed that the change in bone Tregs depends on the absorption rate constants of butyrate and migration rate constant of Tregs. At present, the experimental data to evaluate absorption and migration rate constants are not available for all of the species. We calculated these parameters based on the homeostatic distribution of butyrate and Tregs in mouse models. These parameters may vary between mouse and human models. In Figure 7, we observed a decrease of output bone Tregs with increase of parameter A_B12. In contrast, we did not observe the inverse relationship in the global sensitivity analysis (Figure 8).

Figure 7.

Local sensitivity of the parameters for bone Tregs with respect to output bone Tregs at 28 days. Blue bars indicate that the output and the input change in the same direction, and the red bar indicates that the output decreases when the input increases.

Figure 8.

Global sensitivity of the parameters with respect to output bone Tregs at 28 days. The blue and orange bars represent first-order and total-order indices, respectively.

Model Predictions.

We used the model to predict the long-term dynamics of bone Tregs at specified intestinal butyrate concentrations B_1,ss. Figure 9 shows the dynamics of bone Tregs for 100 days with varying intestinal butyrate concentration. Figure 9A mimics the regenerative condition in which the population of Tregs increases with the increase of intestinal butyrate concentration. High butyrate concentration in the intestine can be toxic and cause cell death instead of pharmacological benefits.^53,54 At present, the pharmacologically active threshold of butyrate for differentiating naïve CD4+ T cells to Tregs is not known, and further experiments are needed to evaluate the threshold for toxicity. Figure 9B mimics the pathological condition where reduction in intestinal butyrate concentration (via disease or antibiotic treatment) reduces the bone Tregs. This disrupts the balance between inflammation and anti-inflammation, and the bone marrow may become inflamed. We can use these results to develop clinical applications and regulate the inflammation with controlled changes in intestinal butyrate concentration. Although the reported butyrate concentrations in the literature are in both the micromolar range^23,55–60 and the millimolar range,^41,53,54 we selected the micromolar range of butyrate, since experimental data for biodistribution of butyrate in that range is available.^18,26 Our current model is scalable to study the effect of the millimolar range of butyrate.

Figure 9.

Predicted bone Tregs at different intestinal butyrate concentrations shown in the legends: (A) increases of intestinal butyrate concentration; and (B) decreases of intestinal butyrate concentration from homeostasis.

We used the estimated parameters to predict the fold change of TGF-β and Wnt-10b in the bone marrow. Figure 10 (blue curves) shows the dynamics that yield 3.73-fold increase of TGF-β and 3.2-fold increase of Wnt-10b in the bone marrow after 28 days of continuous simulated butyrate treatment with B_1,ss = 0.18 μM. The values of $X_{T\beta ,m}^{+}=3.75$ and $X_{W,m}^{+}=3.2$ from Table S1 were specified in the Supporting Information. The simulations with continuous simulated butyrate treatment with B_1,ss = −0.11 μM (orange curves in Figure 10) predicted a 0.6-fold decrease of TGF-β and 0.66-fold decrease of Wnt-10b in the bone marrow. The value of $X_{T\beta ,m}^{-}=0.44$ from Table S2 was specified in the Supporting Information. $X_{W,m}^{-}$ was not specified. Our model can predict the fold change of TGF-β and Wnt-10b for the butyrate dose with an upper limit of 0.2 μM intestinal butyrate concentration to low knockout. If experimental results for higher intestinal butyrate concentration (>0.2 μM) become available, we can use them as input parameters to perform simultaneous iterative curve fitting to estimate the parameters for the higher range.

Figure 10.

Predicted fold changes of (A) TGF-β and (B) Wnt-10b in the bone marrow at different intestinal butyrate concentrations shown in the legends.

Contributions of Butyrate and Wnt-10b in Bone Metabolism.

We investigated the impacts of butyrate and Wnt-10b on bone metabolism using the integrated model, that is, our model developed here connected to our previous detailed bone compartment model for bone metabolism.⁴² Tyagi et al.¹⁸ reported 36.6% increase in trabecular bone volume with LGG treatment for a 3.2-fold change in Wnt-10b by 28 days. The baseline value of trabecular bone volume in the integrated model is 100% at the beginning and end of each cycle without butyrate. Increases due to net bone formation are added to the 100% baseline in the calculation of the relative bone volume percent. We simulated the integrated model without (i.e., B_1,ss = 0) and with gut butyrate concentration set to B_1,ss = 0.18 μM to simulate probiotic LGG treatment for two 14-day bone remodeling cycles. Our results showed 1.95% increase in trabecular bone volume (Figure 11A orange curve) corresponding to the dynamic Wnt-10b profile that reached 3.2-fold change after 28 days (Figure 10B blue curve and Figure S2). These results were well below the experimental 36.6% increase. We next simulated the population dynamics of bone cells—osteoclasts, preosteoblasts, and osteoblasts (eqs S26–S28)—to explore the discrepancy further. Our results showed a slight increase in osteoblasts and small decrease in osteoclasts. The ratios of osteoblasts to osteoclasts and preosteoblasts to osteoblasts were nearly equal between the without and with butyrate simulation cases (Figure 11B–E). Tyagi et al.¹⁸ reported 11.11% increase in osteoblasts with LGG supplement and 25% increase in osteoblasts with butyrate treatment. They did not observe any significant changes in osteoclasts. In contrast, Lucas et al.¹⁶ reported 19.55% reduction in osteoclasts and no significant change in osteoblasts. Another experimental observation showed increased proliferation, differentiation, and maturation of osteoblasts and inhibition of osteoclasts’ proliferation due to LGG and butyrate.⁵⁵ Some of these variations are certainly explainable by differences in experimental protocols and treatments and characteristics of animal models (e.g., age, sex, strain, and vendor) between studies. However, our model was parametrized heavily from the data from Tyagi et al.,¹⁸ so the discrepancy was striking. Then, we reflected on the mechanisms embedded in the integrated model. These mechanisms only allow immune-mediated indirect effects of butyrate on the bone compartment by modulating TGF-β and Wnt-10b.

Figure 11.

Effects of butyrate and Wnt-10b in bone metabolism: (A) relative change in trabecular bone volume, (B) ratio of the cumulative numbers (area under the curve) of osteoblasts to osteoclasts, (C) ratio of the cumulative numbers of preosteoblasts to osteoblasts, and changes in cumulative numbers of (D) osteoblasts and (E) osteoclasts from baseline. The solid circle represents the data from Tyagi et al.¹⁸

We hypothesized that butyrate can directly impact osteoblasts and osteoclasts in the bone metabolism to increase the change in trabecular bone volume. We conducted an in silico experiment to determine possible direct mechanistic impacts of butyrate on bone metabolism. We defined a set of nine mathematical cases that modify the bone metabolism model term-by-term. The cases we considered were baseline condition without butyrate (case 1), baseline condition with butyrate where butyrate indirectly affects bone formation via Wnt-10b (case 2), butyrate increases osteoblasts proliferation (case 3), butyrate increases osteoblasts differentiation (case 4), butyrate increases Wnt-10b mediated osteoblasts differentiation (case 5), butyrate increases osteoclasts death rate (case 6), butyrate inhibits osteoclasts differentiation (case 7), butyrate increases osteoblasts bone formation rate (case 8), and butyrate increases osteoclasts bone resorption rate (case 9). The modified equations for the cases, the corresponding estimated parameters, and the resulting bone cell population dynamics are in the Supporting Information, Figure S3. Cases 3–7 were developed on top of case 2 (allowing for indirect and direct effects of butyrate). We estimated the parameters of cases 3–9 for 36.6% increase in trabecular bone volume after 28 days. We integrated the area under the curve (AUC) to evaluate the ratio of osteoblasts to osteoclasts (Figure 11B) and the ratio of preosteoblasts to osteoblasts (Figure 11C). The changes in osteoblasts and osteoclasts were calculated relative to baseline in case 1 (Figure 11D,E).

In case 3, we observed increased osteoblasts and slightly decreased osteoclasts. In cases 6 and 7, the change in osteoblasts was modest, but reduction in osteoclasts was significant. These cases were consistent with experimental observations. But in cases 4 and 5, we observed decreases in both osteoblasts and osteoclasts. So, we rejected the hypotheses in these cases. Although in cases 8 and 9, we observed increased trabecular bone volume due to increased formation and resorption rates, the changes in osteoblasts and osteoclasts were not significant. We rejected cases 8 and 9 for lack of mechanistic explanation. Our accepted hypotheses (cases 3, 6, and 7) show direct impacts of butyrate in osteoblasts proliferation, osteoclasts differentiation, and osteoclasts death. These cases have implications on designing new experiments in which these generated hypotheses can be tested in vitro and/or in vivo.

CONCLUSIONS

The mathematical model and its solution described here give insight into the biodistribution of Tregs in the gut–bone axis, fold changes of Wnt-10b in bone marrow, and their contributions to modulating bone formation in response to changes in intestinal butyrate formation. We apply this model to investigate the systemic effect of butyrate and Tregs in the maintenance of bone health. We determined that a mechanism that only allowed butyrate to affect the bone compartment indirectly via modulating the immune system and Wnt-10b was insufficient to predict changes in trabecular bone volume resulting from probiotic treatment to increase intestinal butyrate concentration. We conducted an in silico experiment exploring possible mechanisms by which butyrate may directly affect processes in the bone compartment. Through this, we generated hypotheses that can be tested in future in vitro and/or in vivo experiments: butyrate increases osteoblasts proliferation, reduces osteoclasts differentiation, and increases osteoclasts apoptosis. These testable experimental hypotheses require further investigation. The model and results could be used to develop clinical interventions controlling butyrate dose to regulate the level of inflammation in different compartments. This deeper understanding will allow us to explore the potential of modifying butyrate through dietary interventions as a treatment for different pathological conditions related to bone loss such as estrogen deficiency, hyperthyroidism, rheumatoid arthritis, and vitamin D deficiency.

Biographies

Biographies

Mohammad Aminul Islam received his Ph.D. in Chemical Engineering from Missouri University of Science and Technology in 2019. Before joining University at Buffalo, The State University of New York as a Postdoctoral Associate in Dr. Ford Versypt’s lab in 2021, he worked as Postdoctoral Research Fellow at Oklahoma State University in the same group from 2019. His current research activities include pharmacokinetics and pharmacodynamics modeling for local and systemic immune response, inflammation, and tissue damage and how they modulate the balance between healthy and disease states.

Carley V. Cook started her research career as an undergraduate at Oklahoma State University (OSU), where she earned a BS and MS in chemical engineering. In January 2021, she moved to the University at Buffalo to pursue a Ph.D. in chemical engineering as a Schomburg Fellow. Her research focuses on computational modeling of the immune-bone axis.

Brenda J. Smith is a Regents Professor in the Department of Nutritional Sciences at Oklahoma State University. Her research interests are in understanding how dietary bioactive components affect bone metabolism by regulating immunological responses. More recently, her work has focused on the gut–bone axis as a potential target for the treatment and prevention of osteoporosis and inflammatory musculoskeletal disease.

Ashlee N. Ford Versypt is an associate professor in the Department of Chemical and Biological Engineering at the University at Buffalo, The State University of New York. She is also core faculty in the UB Institute for Computational and Data Sciences. She holds three degrees in chemical engineering: a B.S. from the University of Oklahoma and an M.S. and a Ph.D. from the University of Illinois at Urbana–Champaign. Dr. Ford Versypt leads the Systems Biomedicine and Pharmaceutics Laboratory, which develops multiscale mathematical and computational models to enhance understanding of the mechanisms governing tissue remodeling and damage as a result of diseases and infections and to simulate the treatment of those conditions to improve human health. She has received a number of awards for her research, teaching, and service including the NSF CAREER Award, ASEE Chemical Engineering Division Fahien Award, ASEE Midwest Section Outstanding Service Award, AIChE 35 Under 35, University of Notre Dame Thiele Lectureship Award, the OSU Outstanding Achievement for the Mentorship of Women, and the OSU College of Engineering, Architecture and Technology Excellent Teacher Award.