Author's Accepted Manuscript

Mathematical model of mycobacterium-host interaction describes physiology of persistence Gabriele Pedruzzi, Kanury V.S. Rao, Samrat Chatterjee

www.elsevier.com/locate/yjtbi

PII: DOI: Reference:

S0022-5193(15)00146-0 http://dx.doi.org/10.1016/j.jtbi.2015.03.031 YJTBI8133

To appear in:

Journal of Theoretical Biology

Received date: 5 November 2014 Revised date: 24 March 2015 Accepted date: 25 March 2015 Cite this article as: Gabriele Pedruzzi, Kanury V.S. Rao, Samrat Chatterjee, Mathematical model of mycobacterium-host interaction describes physiology of persistence, Journal of Theoretical Biology, http://dx.doi.org/10.1016/j. jtbi.2015.03.031 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Mathematical model of mycobacterium-host interaction describes physiology of persistence Gabriele Pedruzzia , Kanury VS Raoa,b , Samrat Chatterjeeb,∗ a International

Centre for Genetic Engineering and Biotechnology. Aruna Asaf Ali Marg 110 067, New Delhi, INDIA Health Science and Technology Institute, Drug Discovery Research Centre. Plot No. 496, Phase-III, Udyog Vihar, Gurgaon - 122016, Haryana, INDIA

b Translational

Abstract Despite extensive studies on the interactions between Mycobacterium tuberculosis (M.tb) and macrophages, the mechanism by which pathogen evades anti-microbial responses and establishes persistence within the host cell remains unknown. In this study, we developed a four-dimensional ODE model to describe the dynamics of host-pathogen interactions in the early phase of macrophage infection. The aim was to characterize the role of host cellular regulators such as iron and lipids, in addition to the bactericidal effector molecule Nitric Oxide. Conditions for existence and stability of the equilibrium point were analysed by examining the behaviour of the model through numerical simulations. These computational investigations revealed that it was the ability of pathogen to interfere with iron and lipid homeostatic pathways of the host cell, which ensured a shift in balance towards pathogen survival and persistence. Interestingly, small perturbations in this equilibrium triggered the cell’s bactericidal response, thereby producing an oscillatory dynamic for disease progression. Keywords: host-pathogen interaction, differential equations, Iron, Lipids, Nitric Oxide

1

2 3 4 5 6 7 8 9 10 11

1. Introduction Mycobacterium tuberculosis (M.tb), the causative agent of tuberculosis, is a facultative intracellular obligate human pathogen. It is estimated that one-third of the human population is latently infected with M.tb (WHO, 2010). Through long-standing co-evolution with its mammalian host, M.tb has evolved different strategies to invade and survive within macrophages. During the initial phase of infection, the airborne M.tb is phagocytized by the alveolar macrophages where it resides in endosomes, and the virulent strains of the bacteria have the capability to avert endo-lysosomal fusion (Vergne I, 2004). Mycobacteria also subverts the host metabolic machinery and reshapes the transcriptional landscape profoundly (Neyrolles O, 2014). These strategies allow the conversion of the bactericidal macrophage environment into a nutrient rich reservoir for M.tb survival (Vijay S, 2014), facilitating long-term persistence in the ∗ Corresponding

author Email addresses: [email protected] (Gabriele Pedruzzi), [email protected] (Kanury VS Rao), [email protected] (Samrat Chatterjee) Preprint submitted to Journal of Theoretical Biology April 7, 2015

12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

host. Several mathematical and computational approaches have been proposed to complement experimental studies in the field of tuberculosis research (Kirschner D, 2009). These are often targeted towards the pathogen-host interaction in an attempt to describe either the dynamics of infection (Antia R, 1996), or its evolution into the granuloma (Nicholas A, 2013). Attempts to rationalize and predict the cell-mediated immune regulation (Wigginton J, 2001) during M.tb infection are also ongoing. Computational characterizations of latency and persistence have been extensively reviewed (Herbert W, 2000; Stewart G, 2003), and mathematical representations were proposed too (Magombedze G, 2012; Zhilan F, 2000). Interplay between the cytokines interferon-γ (IFN-γ), interleukins 10 and 12 and tumor necrosis factor-α (TNF-α), is now known to play an important role in granuloma formation and stabilization of the infection (Tufariello J, 2003). Within macrophages however, host effectors such as iron, lipids, and nitric oxide serve as important determinants of fate of the intracellular bacteria. Therefore, the competitive interplay between host and pathogen in regulating levels of these intermediates is likely to play an important role in determining the outcome of infection. In light of this, we developed a model to describe the physiological determinants of host and bacterial cell population dynamics during the early phase of infection.

28

29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57

1.1. Biological background The model that we proposed was essentially a four variable ODE model that focused on the reciprocal dynamics of bacterial density with host iron homeostasis, lipid content, and Nitric Oxide (N) biosynthesis. Here we considered iron as an element of crucial importance for the survival, proliferation and functioning of both host and the pathogen. During bacterial infection, the availability of iron positively contributes to the growth of the bacteria (Doherty CP, 2007). Thus, the nutritive strategy of the host is to sequester iron from the plasma, through increased uptake and decreased export of the metal by the macrophages. This is achieved through a hepcidin autocrine loop on macrophages that is activated in response to infection (Weiss G, 2008). Hepcidin induces an increase of transferrin receptors (TfR) synthesis and down regulation of the major iron exporter ferroportin (Fpn1), through iron regulatory proteins (IRP1 and IRP2) (Weiss G, 2008). This mechanism produces a net increase in macrophage intracellular iron content (Appelberg R, 2006), which is a detrimental strategy for the host. Indeed the pathogen has evolved in competition with the host, to subvert and exploit the intracellular environment. M.tb acquires iron by synthesizing iron-binding molecules (siderophores) and recruiting to the phagosome the host iron-transport proteins. The siderophores extract iron from transferrin and shuttle it into the pathogen (Doherty CP, 2007; Boelaert J, 2007). Besides the growth enhancing effect of iron, we also included in our model the physiological co-regulation between N, iron and lipid content. Iron and nitric oxide mutually interact and regulate each other. Iron overload negatively regulates N production by controlling it at the transcriptional level (Weiss G, 1994). On the other hand, N biosynthesis affects cellular iron homeostasis too. It induces an increase of Fpn1 expression, which then induces an increased iron efflux (Kim S, 2003; Nairz M, 2013). Our model also considered the role of lipids and the capacity of M.tb to divert the host glycolytic pathway and citric acid cycle (TCA) to induce lipid bodies accumulation in the macrophage (Singh V, 2012; Merhotra P, 2014). Lipid bodies serve both as a nutritional source (Singh V, 2012) for the bacilli, and as a trojan horse to impair innate immune responses of the macrophage by inhibiting the N production (Huang A, 1999; Yang X, 1994). A supportive line of evidence 1

58 59 60 61 62 63 64 65 66

for building the network of interaction between variables comes from the work from Kram et al. (Kraml PJ, 2005) who showed that increased iron content inside infected macrophages, due to the autocrine effect of hepcidin (Weiss G, 2008; Doherty CP, 2007; Appelberg R, 2006), correlated with oxidization of low density lipoproteins (LDL) and increased cellular cholesterol accumulation (Kraml PJ, 2005). Notably, a high lipid burden is a distinctive histological determinant of foamy macrophages, which are found in the lung granulomas of infected individuals. Interestingly, a similar dynamics of lipid accumulation has also been reported in in vitro models involving macrophages infected with M.tb (Merhotra P, 2014). The complete network of interaction among the variables of our model is graphically summarized in Figure 1.

67

Figure 1: Logic diagram representing the interaction between variables and functional parameters involved in our model. The four coloured box correspond to the model’s variables and the interaction between them are represented with arrows, along with the respective mathematical terms used in the model

68

69 70 71 72

2. Aim of the study By taking the context of a host cell population during the early phase of infection, we aimed to describe the role of iron homeostasis, lipids metabolism and the innate immune response, in terms of Nitric Oxide. Our goal was to probe how these factors interacted, and identify which critical elements either favour or hamper the establishment of persistent infection. 2

73 74

2.1. Assumptions for the model For this mathematical model a series of assumptions were made, which are as follows:

75

76 77

78 79

80 81 82 83

84 85

86 87 88

89 90

91

92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112

• Macrophages were the only host cells considered in the present model. Other cellular interactions that are known to occur in a granuloma were not taken into account. • The growth of mycobacteria was modelled through a logistic equation with a fixed growth rate as assumed by previous work (Ray JCJ, 2008). • The model we proposed contains a minimalist description of lipid metabolism and iron trafficking and does not reach the level of molecular level detail that exists in the biological systems. Nonetheless, it did provide an approximation of the overall dynamics of the system from both a qualitative and quantitative perspective. • The amount of intracellular iron was not compartmentalised, but considered to be entirely available to mycobacteria and linearly interacting with N and lipids. • We did not discriminate between different classes of lipids. Despite the functional and metabolic heterogeneity of lipid molecules, the overall lipid content was accounted within the same variable. • To simplify the innate immunity scenario, we included in our model only Nitric Oxide (N), which was considered as the sole effective weapon against infection 3. Model description In our four-dimensional model, the first variable B(t) represented the bacterial load in a population of cells, which included both infected and resting macrophages. The variables Fe(t), L(t) and N(t) respectively represented the intracellular concentration of iron, lipid content, and the level of Nitric Oxide produced inside the cell in response to infection. The first equation was the backbone of the model and described the intracellular growth of mycobacteria. M.tb growth followed a logistic curve with a fixed growth rate (α1 ), whose value was taken from published work (Vijay S, 2014; Gil W, 2009; Ray JCJ, 2008). During early stages of infection, which naturally occur at a low dose (Saini D, 2012), bacilli are confined in the infected macrophages. Our model described a population of cells that accounted for both infected and uninfected macrophages. Thus, to be representative of the average level of infection in the entire population, the carrying capacity (β1 ) (Ray JCJ, 2008) was adjusted to 5. Iron enhances the growth of bacteria, this effect was modelled with (λ1 Fe) and the microbicidal effect due to N appeared as (-λ3 N). We parametrically defined bacteria to be very susceptible to even low concentration of N, in agreement with published work (Herbst S, 2011; Connelly L, 2001). The second equation was a minimal version of iron homeostasis. It accounted for a constant input of iron inside the cell (Λ1 ), balanced by (-δ1 Fe), the iron consumption and export. The effect of iron withholding was included as an input linearly dependent on bacterial concentration in the form of (γ1 B). The equation took into account a negative contribution from N (-λ4 N), due to transcriptional regulation on Fpn1 expression. In the third equation, similarly as before, there was a constant input (Λ2 ) that described the basal metabolism of lipid and uptake, balanced by (-δ2 L) the rate of efflux and consumption. In the present model we also considered the 3

113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132

capacity of M.tb to divert the TCA cycle and manipulate the host metabolic pathway towards lipid body accumulation (Singh V, 2012; Merhotra P, 2014). Furthermore, we modelled the effects of bacterial infection and iron overload to affect lipids dynamics synergistically in the form (γ2 FeB). This choice was made on the consideration that, only in the presence of infection (B) does an elevated intracellular iron content (Fe) correlate with LDL oxidization and lipid accumulation (Kraml PJ, 2005). In addition to serving as a nutritional source for the bacteria (Singh V, 2012), a high lipid burden also impairs innate immunity by inhibiting the N production (Huang A, 1999; Yang X, 1994). The choice of mass action to include the effect of bacteria on the iron with-holding effect (γ1 B) and on the lipid production (γ2 FeB) was based on the reasoning that these effects would be proportional to the bacterial burden in the macrophage population. Moreover, since the present study anticipated an analytical solution of the system, linear terms were preferred to non-linear ones provided the biological accuracy could be preserved. For the fourth equation, elevated iron counteracted Nitric Oxide induction 3B due to presence of bacteria β2γ+Fe . Similarly as before (γ2 FeB in third equation), we included a combined term where B and Fe act in competition, because this phenomenon is a distinctive feature of the pathophysiology of infection and it is negligible in case of uninfected cells. Therefore, we decided to model the combined effect of Fe and B on L and N. Lastly, the term (-δ3 N) described the negative feedback of N concentration on its own production and (δ5 L) was set to represent the inhibitory effect exerted by high lipid content on N. Combining these assumptions we obtained the following system of equations:

133

134

3.1. ODE equations   dB B + λ1 Fe − λ3 N, = α1 B 1 − dt β1 dFe = Λ1 + γ1 B − δ1 Fe − λ4 N, dt dL = Λ2 + γ2 FeB − δ2 L, dt dN γ3 B = − δ3 N − δ5 L, dt β2 + Fe

135

(1)

with initial conditions B(0) ≥ 0, Fe(0) ≥ 0, L(0) ≥ 0, N(0) ≥ 0.

136 137 138

Before analyzing the system (1), some basic results on the nature of the system solutions were presented below.

139

140 141 142

3.2. Positive invariance By setting X = (B, Fe, L, N)T ∈ R4 and F(X) = [F1 (X), F2 (X), F3 (X), F4 (X)]T , with F : C+ → R4 and F ∈ C ∞ (R4 ), the system (1) became X ∗ = F(X),

143 144

(2)

together with X(0) = X0 ∈ it is easy to check that whenever choosing X(0) ∈ with Xi = 0, for i = 1, 2, 3, 4, then Fi (x) | xi = 0 ≥ 0. Due to the lemma of Nagumo (Nagumo N, 4 R4+ ,

R4+

146

1942) any solution of equation (2) with X0 ∈ R4+ , say X(t) = X(t; X0 ), is such that X(t) ∈ R4+ for all t > 0.

147

4. Mathematical analysis

148

4.1. Equilibrium points and their stability properties

145

149

The system (1) had a single feasible interior steady state E ∗ ≡ (B∗ , Fe∗ , L∗ , N ∗ ) with,

150

L∗ = N∗ = B∗ =

Λ2 + γ2 Fe∗ B∗ , δ2

Λ1 + γ1 B∗ − δ1 Fe∗ λ4

[a(Λ1 − δ1 Fe∗ ) + bΛ2 ](β2 + Fe∗ ) , 1 − (aγ1 + bγ2 Fe∗ )(β2 + Fe∗ )

151 152 153

where a =

δ3 λ4 γ3 ,

b=

δ5 δ2 γ3 .

154

B∗ =

a1 Fe∗2 + a2 Fe∗ + a3 , b1 Fe∗2 + b2 Fe∗ + b3

155 156 157 158

where a1 = −aδ1 , a2 = aΛ1 − aδ1 β2 + bΛ2 , b1 = −bγ2 , b2 = −aγ1 − bγ2 β2 , b3 = 1 − aγ1 β2 .

159 160 161

and solving the system by setting f = − αβ11 , g = α1 −

h = λ1 +

λ3 δ1 λ4 ,

k = − λ3λΛ4 1 we could simplify to obtain:

a3

=

aΛ1 β2 + bΛ2 β2 ,

γ1 λ 3 λ4 ,

162

f B∗2 + gB + hFe∗ + k = 0

(3)

163 164 165

substituting the expression for B∗ , Fe∗ was given by the quintic equation: A1 Fe5 + A2 Fe4 + A3 Fe3 + A4 Fe2 + A5 Fe + A6 = 0

166 167 168

See Appendix A for the full form of the polynomial coefficient for equation (4) 5

(4)

169 170 171 172

4.2. Existence conditions The necessary condition for feasibility of N ∗ was Λ1 + γ1 B∗ > δ1 Fe∗ and we found two more 1 ∗ necessary and sufficient conditions: Λ1 + bΛa 2 > δ1 Fe∗ and β2 +Fe ∗ > aγ1 + bγ2 Fe where for ∗ feasibility of B these two need to have same sign.

173 174

Accordingly to Descartes’ rule of sign we could define a set of sufficient conditions as follow:

175

• aδ1 + bΛ2 > aδ1 β2

176

• aγ1 β2 = 1

177

• α1 > aγ1 β2

178

• ga1 + 2hb2 >

179

• a1 b2 > a2 b1

180

• a22 < 2a1 a3

181 182 183

f (a21 ) b1

+ kb1

Within these conditions, there existed at least one single positive real root for the system and the solutions were feasible. Though the sufficient conditions here defined don’t exclude that the system might have more than one positive solution.

184

185 186 187 188

4.3. Stability conditions Solving for the stability conditions of the system revealed an high analytical barrier. To simplify the analysis we transformed our model into an equivalent dimensionless form by applying the following transformations:

189 190 191

τ = α1 (t), x = βB1 , y = simple form as follows:

δ1 Fe Λ1 ,

z=

δ2 L Λ2 ,

w=

δ3 N γ3 ,

then the system was reduced to a more

dx = x(1 − x) + ξ1 y − ξ2 w, dτ dy = ν1 (1 − y) + ν2 x − ν3 w, dτ dz = ω1 (1 − z) + ω2 xy, dτ dw ζ1 x = − ζ3 w − ζ4 z, dτ 1 + ζ2 y 192 193 194

Definition of the non-dimensional parameters for system (5): ξ1 = αλ11βΛ1 δ11 , ξ2 = αλ1 3βγ1 3δ3 , ν1 = αδ11 , ν2 = ν1 γΛ1 β11 , ν3 = ν1 δγ33Λλ41 , ω1 =

ζ1 =

β 1 δ3 α1 ,

ζ2 =

Λ1 β2 ,

ζ3 =

ζ1 β1 ,

ζ4 = ζ1 δδ52Λγ32 .

195 196

See Appendix B (Stability analysis) for detailed mathematical procedure. 6

(5)

δ2 α1 ,

ω2 = ω1 γδ12 ΛΛ21 ,

Table 1: The fixed set of biologically relevant parameter values with definition. Where [B] : CFU/ml, [Fe] : μM, [L] : μM, [N] : μM

197 198 199

Parameter

Definition

Value

ref.

α1

Growth rate of M.tb

0.025/h

(Vijay S, 2014; Gil W, 2009)

β1

Host carrying capacity

5 CFU/ml

adapted from (Ray JCJ, 2008)

λ1

Effect of Fe on M.tb growth

0.0025 μM−1 CFU/ml h

(Boelaert J, 2007; Doherty CP, 2007)

λ3

Bactericidal effect of N

0.15 μM−1 CFU/ml h

(Herbst S, 2011)

Λ1

Constant input of Fe into the cell

0.001μM/h

estimated here

γ1

Fe with-holding

0.00045/h (CFU/ml)−1

(Appelberg R, 2006; Weiss G, 2008)

δ1

Fe efflux and consumption rate

0.002/h

estimated here

λ4

N negative feedback on IRP1,2

0.0015/h

(Kim S, 2003; Nairz M, 2013)

Λ2

Constant input of L into the cell

0.001μM/h

estimated here

γ2

Synergistic effect of high Fe and B on lipids

0.001μM/h (CFU/ml)−1

(Singh V, 2012; Kraml PJ, 2005)

δ2

Lipids efflux and consumption rate

0.0025/h

estimated here

γ3

Induction of N due to infection

0.004μM/h (CFU/ml)−1

(Herbst S, 2011; Chan ED, 2001)

β2

Threshold for Fe overload inhibitory effect

3μM

(Weiss G, 1994)

δ3

Negative feedback of N concentration

0.006/h

(Connelly L, 2001)

δ5

Lipids inhibitory effect on N

0.001/h

(Yang X, 1994; Huang A, 1999)

From the stability analysis (reported in Appendix A) we found four eigenvalues i ’s, i = 1, 2, 3, 4: 1 = κ1 , 2 = −ν1 , 3 = −ω1 , 4 = J.

200 201

where κ1 = 1 − 2x +

ξ 1 ν2 ν1

and J = −ζ3 + κ4 κκ01 − j5 νκ21 − ζ4 ωκ51 .

202 203 204 205 206 207

One can easily check that 1 < 0 iff κ1 < 0: x >

1 2

+

ξ1 ν2 2ν2 ,

since x =

B β1

we could rewrite

B> + With the set of parameter value defined in Table 1, the quantity 2β11 αλ11γδ11 was negligible, thus we can approximate the condition for stability to be B > 2β11 , where β1 represent the carrying capacity. While 2 and 3 are always negative, and the sufficient conditions for negativity of 4 as follows: 1 2β1 (1

λ1 γ1 α1 δ1 ).

208

• ν2 > ξ1

209

• κ2 < 0

210

• κ5 < 0

211

• j4 > j5 νξ11

212

• κ4 > 0 7

213

and so we could define a sufficient condition for which 4 < 0: − j5 νκ21 − ζ4 ωκ51 < ζ3 − κ4 κκ01 .

214

215 216 217

4.4. Bifurcation analysis In this section we determined the conditions under which Hopf-bifurcation happened in our model using the technique proposed by Wei-Min (Wei-Min L, 1994).

218

219 220 221

Theorem 1. When the parameter δ5 crosses a critical value, say δ∗5 , we have Bn > 0; B1 B2 > B0 B2 and h1 = B1 B2 B3 − B1 2 − B3 2 B0 = 0 and the system (1) enters into Hopfbifurcation around the positive equilibrium that induces oscillations of the bacterial population.

222

223 224

For the mathematical proof of Theorem 1, see Appendix C. A numerical and visual representation of this proof was proposed in Figure 2.

225

0.014

(a)

−0.007

0.013

−0.008

2

−0.009

0.011

δ*5

0.01

−0.012 −12

−10

−8

−6

α

−4

−2

0

−0.013

2

(c)

1

−8

−6

α

−4

−2

0

2 −3

x 10

(d)

4

0.5

0

β

3

−10

2

0.5

β

−12

−3

x 10

1

−0.5 −1

δ*5

−0.011

0.009

1

−0.01

β

β

1

0.012

0.008

(b)

0 −0.5

−1.2

−1

−0.8

−0.6

α3

−0.4

−0.2

0 −3

x 10

−1 −3.2

−3.1

−3

−2.9

α4

−2.8

−2.7

−2.6 −3

x 10

Figure 2: Graphical representation of the eigenvalues for the system (1) i where i = 1a , 2b , 3c , 4d . In (a) and (b) a couple of conjugated complex eigenvalues were represented. The real parts α1,2 crossed the zero axis when δ5 reached the Hopf-bifurcation point δ∗5 . In (c) and (d) two real eigenvalues with no imaginary part.

228

In Figure 2 it was possible to observe the evolution of a couple of eigenvalues, while the parameter δ5 approaches δ∗5 the real part of the couple above crossed the zero axis, thus indicating the presence of Hopf bifurcation.

229

5. Results and simulations

230

5.1. Global Uncertainty analysis

226 227

231 232 233

The accuracy of results from mathematical and computational systems in biology is often complicated by the presence of uncertainties in parameter values. Single-parameter or local sensitivity analyses do not assess uncertainty and sensitivity globally in the system. Global 8

234 235 236 237 238 239 240 241 242 243 244 245 246

Uncertainty (GU) techniques provide a solution in this direction and help to spot the parameter which most critically correlate with the bacterial concentration. The PRCC were measured for each parameter at 1 and 6 months after infection. Latin hypercube sampling (LHS) was used to defined biased-free approximation of the average model output ((Mckay MD , 1979)) and Partial rank correlation coefficients (PRCC) were calculated as illustrated in Marino et al. (Marino S, 2009). The PRCC shows which and how the parameters correlated with the model output at one or six months after infection (Fig. 3). Here we see three peaks (λ1 , γ1 and δ5 ) with significant index absolute value exceeding the Sensitivity Threshold (TS). They positively correlate with the bacterial concentration and represent respectively the growth-enhancing effect of iron (λ1 ), the iron with-holding effect due to infection (γ1 ) and the lipid-mediated inhibition of NO (here δ5 ). Below TS we found two parameters which negatively correlate with bacterial output, the iron consumption rate (δ1 ) and Mtbs growth rate (α). The PRCC can be found in Appendix D (11.1) in the Table 2.

247

Figure 3: Partial Rank Correlation Coefficients (PRCC) were measured at different time point (1 and 6 months after infection) and are plotted in green. A Sensitivity Threshold (TS) of ±0.3 is draw by two red lines. The PRCC profiles at 1 and 6 months after infection appear to mostly overlap. Significance: ∗ : p < 0.025, ∗ ∗ : p < 0.2. 248 249 250 251 252 253 254 255 256

5.2. Local parameter space exploration After the GU analysis we wanted to monitor the effect of local perturbation too on the system dynamics, to show how single modification of single parameter could define changes of the model’s dynamics that might relate to biologically distinctive situations. For this we performed extensive numerical simulations and the results of this exercise are organized in Appendix D (11.2) Table 3. Interestingly we observed that all but three parameter, if varied singularly, could induce the system’s oscillations. The three exceptions included λ1 , β1 and α, the growth-enhancing effect of iron, Mtb’s growth rate and the carrying capacity.

257

258 259 260 261 262 263 264 265

5.3. Basin of attraction A pictorial representation of the topology of basin of attraction was partially defined numerically and is presented in Figure 4. The local stability properties discussed in section 4.3, are not affected by the initial conditions. This exercise aimed to individuate the areas of initial conditions that attract the solutions of the model toward the steady state E ∗ . We considered each couple of variables and varied their initial values in the range i.e. 0 ≤ B(0), Fe(0) ≤ 3, while the steady state values for the remaining two were kept constant (i.e. L∗ , N ∗ ). The areas defined by combination of initial values that attract the solutions toward the stable steady state E ∗ were 9

266 267

exemplified in blue in Figure 4 while the combinations of initial values that drives the solutions away from E ∗ were in red.

268

(b)

2

2

2

NO(0)

3

1 0

1

0

1

2

0

3

0

1

B(0)

2

region depicting unstable solutions stability region of E*

1 0

3

B(0)

0

2

2

1

2

Fe(0)

3

NO(0)

2

NO(0)

3

0

1 0

0

1

3

2

3

(f)

3

1

2

B(0)

3

0

1

(e)

(d)

NO(0)

(c)

3

L(0)

Fe(0)

(a) 3

2

3

Fe(0)

1 0

0

1

L(0)

Figure 4: The areas corresponding to combination of initial values, for each couple of variable, that attracted the solutions toward E ∗ were represented in blue, while the area associated with combinations that drove the solutions away from the stable steady state were in red.

269 270 271 272 273 274 275 276 277 278 279 280 281 282

In Figure 4(a) we observed that stability of the system was dependent on the balance of initial bacterial load and the iron content of the host cell at the beginning of infection. The observed results indicated that under conditions of low initial bacterial load, the high cellular iron content was essential to allow for infection persistence. In Figure 4(b) lipids appeared to be playing a greater role in infection stabilization, since with low lipid content, even when the initial bacterial burden was on the higher end, the system was unable to achieve stability or allow infection persistence. In Figure 4(d) we saw the relation between initial levels of iron and lipids. We could observe a region of instability for low initial levels of these two variables. This result suggested that initial low iron and low lipid content together can be detrimental for persistence of infection. Further, our analysis indicated that high initial N level is extremely detrimental for infection persistence since the system, as shown in Figure 4(c,e,f) was unable to achieve stability, irrespective of the initial bacterial burden, iron or lipid content. The overall analysis pointed towards Nitric Oxide being the most crucial variable that influences infection persistence, having a greater impact when compared to lipid or iron content.

283

10

284

285 286 287

5.4. Model dynamics The dynamics of the model was sensitive to parameter perturbations and behavioral switches could be observed as illustrated in Figure 5. This exemplified a switch between a stable steady state and oscillatory behavior, when γ3 was increased.

288

(b) γ = 0.0043

(a) γ = 0.004

3

3

*

1.5

1.4

E

Lipids

Lipids

1.2 1

0.5 1

0.9

0.8

Iron

0.7

0

3

2

1

1

4

E*

1 0.8

0.9

0.8

0.7

0

Iron

Bacteria

1

2

3

4

Bacteria

(c) γ = 0.0046 3

1

*

E

Lipids

0.9 0.8 0.7 1

0.9

0.8

Iron

0.7

0

1

2

3

4

Bacteria

Figure 5: Phase plane with γ3 as bifurcation parameter. The blue star indicated that the interior equilibrium point E ∗ was stable in (a) and in (b) too, despite the solutions were periodic. In (c) the red star defined the instability of E ∗ and we observed the presence of a Hopf-bifurcation.

289 290 291 292 293 294 295 296 297 298 299 300

As can be seen in the Figure 5(a), the model reached a stable steady state (indicated by the blue star) for γ3 = 0.0040, while a shift to oscillatory behaviour happened for higher value of the parameter (γ3 = 0.0043) (Fig. 5(b)). A further increase, (γ3 = 0.0046) (Fig.5(c)), made the steady state unstable (red star). Slight variation of γ3 , could modify the behaviour and the local stability properties of the model. The presence of a Hopf-bifurcation point at γ3 = 0.0042, as reported in Table 2 (Appendix D), was interesting. A stepwise increase in γ3 (the parameter representing infection induced N production), caused the system to shift from stability to oscillatory behavior. The phenomenon observed here highlighted the nature of the host-pathogen interaction dynamics. The achievement of a certain N threshold seemed to be crucial to shift the system from persistence of infection toward the onset of a competition between host and the pathogen. This competition, mathematically depicted by Hopf-bifurcation, consisted in a dynamical balance in which the host actively defends against bacterial infection.

301

302

303 304 305 306

5.5. Time course predictions The general behaviour of the system related to parameter perturbations that were in agreement with the dynamics expected from literature reports. In a few cases the model revealed high susceptibility to parameter perturbation. A time course representation was depicted to provide a clear representation of the nature of dynamics predicted by the model (Fig.6).

307

11

δ3= 0.052

(a)

1.5

3 2

L

[B,F,L,N]

4

1 4

1 0 0

0.5

1

1.5

2

time (h.)

0

0.7

0.95

0.9

0.85

1

Fe

1 0.5 4

1 0 0

0.5

1

1.5

2

time (h.)

2.5

2

NO

4

x 10

0.9

0.8

1

Fe

δ = 0.046

(c)

3

1.5

3 2

L

[B,F,L,N]

0.8

1.5

2

1 0.5 4

1 0 0

0

δ3= 0.049

(b)

3

4

2

NO

4

x 10

L

[B,F,L,N]

4

2.5

[bacteria] [iron] [lipids] [nitric oxide]

0.5

1

1.5

2

time (h.)

2.5

2

NO

4

x 10

0

0.8

0.7

0.9

1

Fe

Figure 6: Time series and phase-portrait for the model using different values of δ3 (negative feedback of N concentration). In (a) the model rapidly reached the stable steady states E ∗ . In (b), with a lower value of δ3 , the system showed dampening oscillations and seemed to approach E ∗ , but failed to do so. Oscillation with increasing amplitude were triggered on and the solution did not reach a stable point. In (c), an even lower value of δ3 , defined a rapid onset of periodic solutions.

308 309 310 311 312 313 314 315 316 317

We had already observed that a slight change of the parameters induced the system to deviate from a stable steady state, descriptive of persistent or para-symbiotic infection, to oscillations in Figure 5(a-c). Observing the time course predictions, here in Figure 5(a), the bacterial burden settled to approximately 2.12 or, in other words, to nearly half of the carrying capacity of the system (β1 = 5). In this scenario, the bacterial burden asymptotically settled to a stable equilibrium with a negligible production of Nitric Oxide, which was present but only at very low levels. Figure 5(b) showed how a slight decrease of δ3 , representing the negative feedback of N concentration, induced an unstable state characterized by oscillations with increasing amplitude of bacterial load. The oscillatory solution related to fluctuations in bacterial density due to a competition between proliferation and killing of the bacteria.

318

319

320 321

5.6. Single parameter bifurcation analysis

We presented a bifurcation diagram which showed stable steady state for α1 < 0.012 and α1 > 0.02, while for 0.012 ≤ α1 ≤ 0.02 periodic oscillations were observed.

322

12

[Bacteria]

4 2 0 0.005

0.01

0.015

α

1

α

1

α

1

0.02

0.025

0.03

0.02

0.025

0.03

0.02

0.025

0.03

0.02

0.025

0.03

[Fe]

1.5 1 0.5 0.005

0.01

0.015

[Nitric Oxide]

[Lipids]

2 1 0 0.005

0.01

0.015

0.4 0.2 0 0.005

0.01

0.015

α

1

Figure 7: Bifurcation diagram with α1 as bifurcation parameter (M.tb growth rate). For values of α1 < 0.0125 or α1 > 0.021 the solutions of the system converged on the stable point E ∗ . In the range 0.0125 < α1 < 0.021 the system oscillated with increasing amplitude as α1 increased.

323 324 325 326 327 328 329 330

From Figure 7 we could observe that bacteria with significantly different growth rates established a different balance within the host. Thus, bacilli showing medium growth rates caused the system to oscillate, while the ones with either very high or very low growth rates tipped the system toward stability. The observation is often what is seen in biological systems. M.tb infects cells at a low multiplicity of infection, which usually results in dormancy and persistence, a condition reflective of a para-symbiotic relationship. We next investigated this phenomenon further by varying β2 , which defined the threshold of inhibition of N by the iron content. The results were shown in Figure 8. 13

[Bacteria]

10 5 0 0.5

1

1.5

2

β

[Fe]

2

1

1.5

2

β

3.5

4

2.5

3

3.5

4

2.5

3

3.5

4

2.5

3

3.5

4

2

4 [Lipids]

3

1 0 0.5

2 0 0.5

[Nitric Oxide]

2.5

2

1

1.5

2

β

2

0.4 0.2 0 0.5

1

1.5

2

β

2

Figure 8: Bifurcation diagram depicting the effect of varying β2 . For values of β1 > 2.8 the solutions of the system converged on the stable point E ∗ , while for β1 < 2.8 the system oscillated with increasing amplitude as β1 was increased.

331 332 333 334 335 336 337 338 339

Periodic oscillations with increasing amplitude were observed as β2 was increased and a switching point for β2 = 2.8. The range of stability for this parameter was 2.8 < β2 < 3.7 and inside this window the system was stable and reached the steady state E ∗ . Oscillatory behaviour was observed for β2 < 2.8, the amplitude of the oscillation reduced proportionally with β2 . This observation underscored the existence of a range of iron concentration which is able to counter balance the anti-bacterial effect of N levels in host cells. The existence of a defined minima and maxima of iron content for achieving stability highlighted the significance of delicately balanced host parameters that determine the outcome of infection and suggested how seamlessly the bacteria has evolved to ”fit and feed” inside the host machinery.

340

341

342 343 344 345 346 347

5.7. Two parameter bifurcation analysis

To understand the link between the growth rate of pathogen and its effects on host lipid metabolism, we proposed a two-parameter bifurcation diagram in Figure 9. It exemplified the combined effect of different values of α1 , growth rate of the bacilli and the induction of lipid bodies by the pathogen γ2 , in a three-dimensional space that reported the bacterial burden on the z axis. A two dimensional visualization was also depicted to better capture the concept revealed by this graphical representation.

348

14

6

B*

4

2 0.04 0 2

0.02 1.5

1

−3

x 10

0.5

0

γ2

0

α1

−3

2

x 10

5 4

γ

2

1.5

3

1

oscillatory solutions

2

0.5 0

*

colorbar: [B ]

1 0.01

0.02

α1

0.03

0.04

0

Figure 9: Bifurcation diagram with α1 (logistic growth rate of bacteria) and γ2 as bifurcation parameters (lipid induction by Fe and B). The blue area indicated the presence of oscillatory solutions while the color graded area stood for the stable solutions E ∗ . The color grade conveyed the bacterial concentration at the steady state, as function of the two parameters α1 and γ2 . 349 350 351 352 353 354 355 356 357 358 359

Here the yellow to red dotted region represented the portion of the parameter space associated with stability of steady state E ∗ , which allowed persistence of infection. The color graded bar on the right side of the figure conveyed the bacterial burden at E ∗ . On the other side, the region of the graph in blue represented instability of E ∗ and was associated with oscillatory solutions. Interestingly, the host lipid manipulation correlated with growth rate in a nonintuitive way. For slowly proliferating bacteria (α1 < 0.017) the magnitude of lipid induction required for attaining persistence (shift from blue to orange area of the graph) rises much faster as the growth rate increases. For higher values (α1 > 0.017), however, the corresponding values for γ2 at the border between the two regions, inversely correlated with α1 and decreased to zero when the growth rate crossed a threshold α1 = 0.036. Beyond this critical value, the fast growth rate alone could define the onset of persistence of infection even without any lipid induction (γ2 ≈ 0).

360

361

362 363 364 365 366

6. Discussion In the present work was widened the hypothetical boundary of investigation beyond that of a single infected cell. We considered an infected population of cells as it typically is, characterized by significant phenotypical and functional heterogeneity (Mills CD, 2012; Corna G, 2010). Furthermore, the outcome of M.tb infection is also related to the general health condition of the infected host and is thus expected to vary between individuals. The complexity and dynamics 15

367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413

of the infection process cannot be easily recapitulated in a standardized in vitro setting. For these reasons a mathematical model was employed to study the phenomenon of intracellular mycobacterial persistence. We addressed the issue analytically, with the aid of a computational approach, to complement and validate the experimental data. Our model had a single feasible stable steady state E ∗ . The first sufficient condition found for stability in our model confined the bacterial clearance outside the margin of stability of the system. A minimal amount of bacteria B > 2β11 (1 + αλ11γδ11 ) was found to be necessary for the existence and stability. We correlated this feature with the nature of intracellular mycobacterial persistence, which to some extent resembles a para-symbiotic relationship. Recursive infectious episodes and persistence are characteristic of tuberculosis. Latent infections are the most common whereas complete clearance of bacteria by the host is extremely rare. Although persistence was earlier related to bacterial dormancy (Wayne LG, 1994), recent work has shed new light on this aspect. Das et al. (Das B, 2013), have elegantly shown that bone marrow derived stem cells are a reservoir for M.tb. These cells host dormant bacteria and fuel a slow-dynamic persistence mechanism. Further, in a study of bacterial turnover and kinetics within the host, Dannenberg et al, have suggested that bacterial persistence at low numbers may simply reflect a situation where proliferation and killing occur at equal rates in the macrophage (Dannenberg AM, 2003). These perspectives supported a dynamical interpretation of persistent infection in accordance to our model predictions. Interestingly, a recent genomic sequencing study (Ford C, 2011) has defined a similarity in the mutation rate of mycobacteria from cases of active tuberculosis and latency. In this work, the oxidative DNA damage was claimed as the contributor for the single nucleotide mutation burden found in the latency condition. These findings suggested that engagement between host and pathogen indeed involves a finely regulated balance between proliferation and killing of pathogen through reactive oxygen species and N. Taken together with existing literature, our studies proscribed a conceptual framework for interpretation of bacterial persistency as a dynamical process. Recently, Pienaara and Lerm also employed a computational approach to investigate the interplay between host and pathogen. They found that a transient oscillating balance between the interacting entities contributed to different incubation periods in infected subjects (Pienaara E, 2014). A similar dynamical perspective was confirmed by our study. With the present model we only discriminated between persistence (E ∗ ) and host response (oscillatory solutions). We individuated the physiological drivers that can tip the balance between these two distinct biological scenarios. Whether a host response is favourable or not to the host, is beyond the prediction power of this model. At the population level, bacterial persistence in the form of a latent infection is the outcome in roughly 90% of M.tb infections (WHO, 2010). Thus, persistence is an important feature of tuberculosis and our study addressed this feature specifically. In this scenario the infection is confined inside the macrophages and it shows as asymptomatic. This condition in fact resembles a para-symbiotic relationship that help both the pathogen and the host to some extent. The extensive numerical simulations revealed in fact a delicate balance and underscored that a perturbation in several parameters produced behavioral switches between steady state and Hopf-bifurcation. These results were consistent with the hypothesis that persistence of tuberculosis infection strikes out a delicate balance within the host. Eventually only about 5-10% of the infected individual progress to active disease (Stewart G, 2003). Once the analytical conditions for local stability were found, our numerical simulations could be oriented to visualize the topology of basin of attraction. This exercise, depicted in Figure 4, suggested that the initial concentration of Nitric Oxide was the crucial element determining the 16

414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460

balance between persistence, or host-mediated killing of the pathogen. A visualization of the dynamics predicted by the model was presented in Figure 6. The parameter chosen to exemplify the behavioural switches between steady state and oscillatory solutions was δ3 , the feedback of N concentration on its own production. The steady state (Fig.6(a)), indicated that the shift between bacterial persistence within host and the onset of a host-initiated bactericidal response was critically sensitive to parameter perturbations. Persistence was represented as delicate balance, involving host physiology and pathogen in a para-symbiotic relationship. With decreasing δ3 the host response was triggered and oscillations started to appear with an increasing amplitude (Fig.6(b)). This different situation related to a progressive evolution of the competition between host and pathogen. Our results ultimately suggested that the kinetics of this progression depends on the negative feedback of N concentration (δ3 ). In fact, in Figure 6(c), we observed the onset of a host response was faster if δ3 was further decreased. Interestingly, N itself exerts a biphasic effect on the transcription of iNOS. Low concentrations of N can in fact activate NF-κ B and up-regulate iNOS while high concentrations have the opposite effect (negative feedback), which may help prevent N overproduction (Umansky V, 1998; Connelly L, 2001). Our model suggested that the reduction of the self inhibitory loop of N aids the host response despite the establishment of persistence. A recent study (Bhat KH, 2013) has revealed that M.tb-dependent inhibition of N production by macrophages is mediated through secretion of the PE/PPE family of proteins. While the mechanism of this inhibition remains to be fully understood the findings, however, are consistent with our model. We observed that high values of δ3 , that is an increase of the N negative feedback due to pathogen-mediated perturbation of host cell signalling, assisted the establishment of persistence. These result indicated how the dynamics of disease progression is driven by crosstalk between host and pathogen, and how slight modifications in Nitric Oxide metabolism profoundly affected the outcome. A related objective of our study was also to investigate the possible strategies, or bottlenecks, underlying the establishment of persistent infection. We wanted to know the physiological processes that contribute to restraining the infection, and also how perturbations in the parameter set can affect the overall output of the model. From the Global Sensitivity (GU) analysis we observed that few parameter most strongly correlate with the bacterial density. We observed that a λ1 (the growth-enhancing effect of iron) and the iron efflux rate (δ1 ) correlated with model outcome as their Partial Rank Correlation Coefficient (PRCC) exceed the Sensitivity Threshold (TS) defined. In particular we observed the PRCC of λ1 is higher at 1 month rather than 6 months after infection. This result suggested a prominent role of iron metabolism in initial phase to establish infection. Another element that positively correlate to bacterial was γ1 (iron withholding effect) which also supported the persistence of infection. From a biological standpoint, this parameter directly defined an increase in iron uptake and storage by the macrophage in response to infection. Withholding iron from invading pathogens represents a critical host defense strategy. In the case of M.tb infection however, sequestration of iron leads to the provision of an ideal cellular iron source for this pathogen (Collins H, 2008). The predictions of our model agreed with published experimental results (Boelaert J, 2007), as macrophage iron overload has been shown to worsen the outcome of tuberculosis in humans. Moreover, anemia is a known risk factor for tuberculosis and high prevalence of anemia is found among tuberculosis patients and it correlates with an increased risk of death (Isanaka S, 2012). However, it is still uncertain if anemia represents an a priori risk factor for TB, or the consequence of a chronic infection. As Weiss has tried to clarify (Weiss G, 2008), anemia is most frequently found in hospitalized patients suffering from chronic inflammatory disorders. The diversion of the metal from the circulation to the cellular storage compartments results, 17

461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507

at the same time, in hypoferremia (anemia) and hyperferritinemia (increased storage of iron). Thus, while a person chronically infected with TB might have blood-anemia, this does not exclude the possibility that the iron content inside macrophages could be elevated and, thereby, fuel the chronic infection. Although our model do not confirm such a hypothesis, the predictions nonetheless, endorsed iron withholding as a factor that negatively affects disease outcome. The parameter γ3 , or the rate at which the infection can activate the innate response and N production, was critical and may vary considerably depending on the infecting strain, as well as the individual being infected. Mycobacteria adopt different evolutionary strategies to establish within the host, and it is known to hamper or minimize the production of N, though inhibition of IFN-γ (Sibley LD, 1988). Our model predicted that a higher rate of induction of Nitric Oxide, representative of cells stimulated by IFN-γ or activated macrophages, favoured the establishment of oscillatory behaviour and triggered a host’s response. This finding was in agreement with previous studies which showed that the sensitivity of macrophages to infection and their ability to fight depends on their activation status and N production. Pre-exposure to IFN-γ can improve the bacterial clearance by consequently increasing the production of reactive nitrogen intermediates (Denis M, 1991; Flesch I, 1987). Some yet undefined threshold of activation is necessary for the induction of Nitric Oxide by infection and this can vary between individuals and in different conditions, and especially during a different phase of infection. Our results suggested that this inhibitory threshold might be tracked back to the iron overload, which plays an important role in regulating the production of Nitric Oxide. Iron overload, can be primarily interpreted as a consequence of infection itself, but indeed represented a tool exploited by the pathogen to impair the production of N as shown by Figure 8. The balance that allows establishment of persistence must then be a subtle compromise between the rate of N production, the iron content and the proliferation rate of the bacilli. Another interesting feature of the model was indeed the growth rate of bacteria and the bifurcation diagram proposed in Figure 7. Stability and conditions that favour persistence were found for values of 0.012 < α1 > 0.02 while Hopf bifurcation happened in the range 0.012 ≤ α1 ≤ 0.02. Interestingly, from the GU analysis we observed that the growth rate α negatively correlated with the bacterial concentration, which is a counterintuitive result if we imagine that fast growing bacteria might be fitter to infect the host. By varying bacterial growth rates, we next sought to understand what factors defined the critical balance during the early phase of infection. For this reason we decided to further study the effect of different growth rate and lipid induction γ2 . Our numerical simulations suggested that pathogen-mediated manipulation of lipid metabolism and iron homeostasis constituted an interwoven strategy for subverting the host response. Particularly narrow stability ranges revealed that lipid metabolism exerts a tight control on the overall behaviour of the model. In the case of γ2 , the stability range for this parameter was limited to a narrow range of values, and minor changes could dramatically affect the system dynamics. The biologically related situation that was mimicked represents the bottleneck of the system, and an evolutionary strategy for M.tb to overwhelm the host resistance. With the GU analysis we also identified that the parameter for the luipid induced NO inhibition δ5 affect the global susceptibility of the system. From the parameter space esploration (Appendix D), we then saw that if considered togheter γ2 (the lipid accumulation effect induced by M.tb) and δ5 , the model stability was restricted to combination of narrow ranges of these parameters, and the behaviour was extremely sensitive to slight parameter perturbation. In such conditions, persistence evolved as a delicate balance between host and pathogen interplay. These results lead us to the interpretation that the host cell lipid metabolism manipulations are a strategy adopted 18

508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523

by M.tb to hamper the nitric oxide production, thus obtaining immune evasion. Since it is known that different strains of M.tb have different properties, and have resourced different adaptation strategies to survive within host, we were interested to see how the growth rate and the rate of lipid accumulation correlate. In Figure 8 we identified two distinct regions, in yellow to red the combinations of parameters α1 and γ2 , which related to the stable steady state. The blue areas, on the other hand, defined the parameter combinations that induced the periodic oscillations. The lower blue portion indicates that the host mounted a response against the pathogen. Interestingly we observed that the correlation between these two parameter varied unexpectedly. The role of lipid induction (γ2 ) appeared to be of extreme importance and to directly paralleled the growth rate, only for bacteria that exhibited lower growth rates (0.01 < α1 < 0.017). Beyond some critical value (α1 > 0.017) the two parameters inversely correlated, and the magnitude of lipid induction that was necessary to obtain persistence decreased as the growth rate increased. These results highlighted that bacteria with different growth kinetics may have evolved modified strategies to subvert the host response and attain a persistent occupancy of the cell. Notably, we can extrapolate that the fast growing bacteria need not rely on recalibrating lipid metabolism to hijack the host, although this appeared to be the favoured strategy for slower replicating bacteria.

524

525

7. Conclusions

542

The present model was built upon multiple sources of data, mostly from in vitro experiments described in the literature. Instead of targeting a single molecular pathway, we aimed to model the physiology of host-pathogen interaction, and provide a mechanistic interpretation of persistence. The present system had one interior stable steady state E ∗ (B∗ , F ∗ , L∗ , N ∗ ) for which existence and stability conditions were found. The GU analysis and numerical evaluation of the model revealed the presence of interweaving roles for lipid metabolism and iron homeostasis in determining the persistence of infection. Here we identified a strong sensitivity of the model toward initial levels of N, rather than lipids or iron. We observed the presence of Hopf-bifurcation occurring for several parameters after perturbation of the parameter value. On the other hand, persistence of infection appeared to result from a fine balance between multiple factors. We also further evaluated the manipulation of host cell lipid as a critical strategy for the establishment of persistence of infection. Our model suggested that the effect of iron overload and lipid accumulation synergistically affected the fitness of the system in terms of host cell response and ability to restrain the infection. Furthermore, our numerical simulation and ODE analysis suggested a link between bacterial growth rate and lipid accumulation following infection. A plausible virulence strategy during early phase of infection is suggested in terms of an inverse correlation between the growth rate of pathogen and the magnitude of lipid manipulation.

543

8. Appendix A

544

8.1. Extended form of the polynomial coefficients for equation (4)

526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541

545 546 547

A1 = hb21 , A2 = f a21 + ga1 b1 + 2hb1 b2 + kb21 , A3 = 2 f a1 a2 + g(a1 b2 + a2 b1 ) + h(b22 + 2b1 b3 ) + 2kb1 b2 , A4 = f (a22 + 2a1 a3 ) + g(a1 b3 + a2 b2 + a3 b1 ) + 2hb2 b3 + k(b22 + 2b1 b3 ), A5 = 2 f a2 a3 + g(a2 b3 + a3 b2 ) + hb23 + 2kb2 b3 , A6 = f a23 + ga3 b3 + kb23 .

548

19

549

9. Appendix B

550

9.1. Stability analysis

551 552

The jacobian matrix of the system (5) around any arbitrary point (B, Fe, L, N) or 1 zΛ2 wγ3 (xβ1 , yΛ δ2 , δ2 , δ3 ) was given by ⎛ ⎜⎜⎜ 1 − 2x ⎜⎜⎜ ν 2 J ≡ ⎜⎜⎜⎜ ⎜⎜⎝ ω2 y j4

ξ1 −ν1 ω2 x − j5

⎞ 0 −ξ2 ⎟⎟ ⎟ 0 −ν3 ⎟⎟⎟⎟ ⎟, −ω1 0 ⎟⎟⎟⎟⎠ −ζ4 −ζ3

(6)

553 554 555

where j4 =

ζ1 1+ζ2 y ,

j5 =

ζ1 ζ2 x . (1+ζ2 y)2

556 557 558

The four eigenvalues εi , i = 1, 2, ∗3, 4 corresponding to the jacobian matrix of the dimension∗ ∗ less system (5) around E ∗ ≡ (x∗ β1 , y δΛ2 1 , z δΛ2 2 , wδ3γ3 ) were given by the equation: ε4 + B3 ε3 + B2 ε2 + B1 ε + B0 = 0

559 560 561 562 563

(7)

where: B3 = −1 + 2x + ν1 + ω1 + ζ3 , B2 = ω1 ζ3 − ω1 − ζ3 + ν1 (ω1 + ζ3 − 1) + 2x(ν1 + ω1 + ζ3 ) − j5 ν3 − ν2 ξ1 + j4 ξ2 , B1 = ν1 [2x(ω1 + ζ3 ) + ω1 (ζ3 − 1) − ζ3 ] − ν2 [ξ1 (ω1 + ζ3 ) + ξ2 j5 ] − ν3 [ j5 (2x − 1 + ω1 ) + xω2 ζ4 ] + j4 [ξ2 (ν1 + ω1 ) + ξ1 ν3 ] + ω1 ζ3 (2x − 1) − ω2 yξ2 ζ4 , B0 = ω2 ζ4 [−ν3 (2x2 − x + yξ1 ) − ξ2 (ν2 x + ν1 y)] + ω1 ζ3 [ν1 (2x − 1) − ν2 ξ1 ] + ω1 [−ν3 j5 (2x − 1) + j4 (ν1 ξ2 + ν3 ξ1 ) − ν2 j5 ξ2 ]

564 565 566 567 568 569 570 571

To find the conditions that satisfy Routh-Hurwitz (Hurwitz A, 1895) criterion for stability of a four degree polynomial (Bn > 0, B3 B2 > B1 B4 , B3 B2 B1 > B4 B21 + B3 B0 ) still posed a great deal of analytical difficulties. So to find the eigenvalues explicitly, we transformed the matrix J to a lower triangular matrix J . Before making the transformation we assume here, the matrix J to be a real square matrix. We then performed a series of row and column operations, which are listed in detail below in subsequent subsection of Appendix B (9.2). As a result of these operations J was now converted to a lower triangular matrix J , that preserved the eigenvalues over the reals. ⎛ ⎜⎜⎜ κ1 ⎜⎜⎜ ν − ξ 1 J ≡ ⎜⎜⎜⎜ 2 ⎜⎜⎝ κ3 j6

572 573

0 −ν1 ω2 x j7

0 0 −ω1 −ζ4 − κ5

0 0 0 J

⎞ ⎟⎟⎟ ⎟⎟⎟ ⎟⎟⎟ , ⎟⎟⎟ ⎠

(8)

Where κ1 = 1 − 2x + ξν1 ν1 2 and J = −ζ3 + κ4 κκ01 − j5 νκ21 − ζ4 ωκ51 . The reader can refer to the subsection 9.4 of Appendix B for all the extended form of the other terms present in matrix (8).

574 575 576

That allowed us to identify eigenvalues directly and define stability conditions. The characteristic roots of the matrix i ’s, i = 1, 2, 3, 4 were given by the elements of the diagonal

577 578

1 = κ1 , 2 = −ν1 , 3 = −ω1 , 4 = J.

579

20

580 581 582

9.2. Row column operation used in section 4.3 R1 → R1 + R2 ( νξ11 ), C1 → C1 + C2 ( νξ11 ); C4 → C4 + C1 ( κκ01 ), R4 → R4 + R1 ( κκ01 ), κ2 C4 → C4 + C2 ( nu ), R4 → R4 + R2 ( νκ21 ); C4 → C4 + C3 ( ωκ51 ), R4 → R4 + R3 ( ωκ51 ). 1

583

584 585 586

9.3. Extended form of the terms present in simplified notation in matrix (8). k0 = ξ2 + νν3 ξ1 1 , κ2 = −ν3 + κκ01 (ν2 − ξ1 ) κ3 = ω2 (y + x νξ11 ) κ4 = j4 − j5 νξ11 k5 = κ0 κκ31 + xω2 νκ21 j6 = κ4 + κ0 + νκ21 (ν2 − ξ1 ) + κ3 ωκ51 , j7 = − j5 − κ2 + ω2 x ωκ51

587

588

589 590

10. Appendix C Proof of Theorem 1. Assuming Bn 1 > 0 and B1 B2 > B0 B2 , the necessary and sufficient conditions for Hopf-bifurcation to occur at δ5 = δ5 ∗ are that:

591 592

(i) g(δ∗5 ) ≡ B1 (δ∗5 )B2 (δ∗5 )B3 (δ∗5 ) − B1 (δ∗5 )2 − B3 (δ∗5 )2 B0 (δ∗5 ) = 0

593 594

(ii)

d ∗ dδ5 Re(ε(δ5 ))δ5 =δ5

0

595

596

Let at δ5 = δ5 ∗ where δ5 =

ζ4 δ2 γ3 ζ1 Λ2

, h1 = 0 then from the characteristic equation we have:

597

ε4 + B3 ε3 + B2 ε2 + B1 ε + 598 599 600

B2 B1 B2 − 12 B3 B3

=0

(9)

Since the determinant of the quartic characteristic polynomial (evaluated numerically with the parameter set given in Table 1 with δ5 = δ5 ∗ ) Δ < 0 then the equation has two real roots and two complex conjugate roots (Rees E, 1992) of the form:

601 602 603

ε1 (δ5 ) = α + iβ(δ5 ) ε2 (δ5 ) = α − iβ(δ5 )

604 605 606 607

while the other two roots are less than zero around the critical point as conditions for local stability hold: ε3 (δ5 ) = −0.0027 ε4 (δ5 ) = −0.0007

608 609 610

Next to verify transversality condition we substituted ε j (δ5 ) = α1 + iβ1 (δ5 ) into (9) and calculating the derivative on both sides we obtained:

611 612 613

α 1 (δ5 )K(δ5 ) − β 1 (δ5 )L(δ5 ) + M(δ5 ) =

α1 (δ5 )L(δ5 ) + β 1 (δ5 )K(δ5 ) + N(δ5 ) = 0

0

614 615

where:

616 617 618

K(δ5 ) = 4α3 − 12αβ2 + 3B3 (α2 − β2 ) + 2B2 α + B1 L(δ5 ) = 12α2 β + 6B3 αβ + 2B2 β − 4β3 21

619 620

M(δ5 ) = B 1 α + B 1 ( BB23 − N(δ5 ) = B 1 β

2B1 ) B3 2

621 622

Assuming K M + LN  0 we have:

623 624

d ∗ dδ5 Re(ε j (δ5 ))δ5 =δ5

=

K M+LN K 2 +L2

 0, for δ5 = δ∗5

625 626 627

Therefore the transversality condition holds. This implies that a Hopf-bifurcation occurs at δ5 = δ∗5 and is non-degenerate where K M + LN  0.

628

629

11. Appendix D

630

11.1. Global Sensitivity Analisys and PRCC values.

Table 2: Partially ranked correlation coefficients (PRCC) of each parameter in respect to the bacterial growth. Only eleven (∗ ) parameters showed a significant PRCC p ¡ 0.05 (∗ ∗: p ¡02). The PRCC value reported in the Table were measured 1 or 6 months after infection.

Parameter α β2 λ1 λ3 λ4 Λ1 Λ2 γ1 γ2 γ3 δ1 δ2 δ3 δ5 dummy

PRCC 1 mo.

PRCC 6 mo.

-0.4040 * 0.1793 * 0.5559 * 0.0125 * 0.0510 0.2520 ** 0.1006 * 0.4706 * 0.2036 * -0.2403 * -0.6236 * 0.2549 * 0.0079 0.3721 * 0.0118

0.4747 * 0.2119 * 0.4618 * -0.0610 ** 0.0184 0.2340 * 0.0605 ** 0.4484 * 0.2134 * -0.1975 * 0.6340 * -0.2518 * -0.0386 0.3305 * 0.0215

22

631

11.2. Parameter space exploration for biologically relevant values

Table 3:

632

Parameter

Range

Existence of stable E*

Existence of periodic sol.

α1

.01 ≤ α1 ≤ .04

.01 < α1 < .012 .02 < α1 < .04

.012 ≤ α1 ≤ .02

β1

1 ≤ β1 ≤ 5

always exist

Do not exist

λ1

.001 ≤ λ1 ≤ .005

always exist

Do not exist

λ3

.1 ≤ λ3 ≤ .5

.1 ≤ λ3 < .17

.17 ≤ λ3

Λ1

.0005 ≤ Λ1 ≤ .002

.0008 < Λ1 ≤ .0013

Λ1 ≤ .0008

γ1

.0001 ≤ γ1 ≤ .001

.00036 < γ1 ≤ .0005

γ1 ≤ .00036

δ1

.001 ≤ δ1 ≤ .005

.0018 ≤ δ1 < .0022

.0022 ≤ δ1

λ4

.001 ≤ λ4 ≤ .005

.001 ≤ λ4 < .0025

.0025 ≤ λ4

Λ2

.0005 ≤ Λ2 ≤ .005

.007 < Λ2 ≤ .0024

Λ2 ≤ .0007

γ2

.0001 ≤ γ2 ≤ .002

.0008 < γ2 ≤ .0011

γ2 ≤ .0008

δ2

.0005 ≤ δ2 ≤ .005

.0028 < δ2 ≤ .0034

.0028 ≤ δ2

γ3

.001 ≤ γ3 ≤ .01

.0035 ≤ γ3 < .0042

.0042 ≤ γ3 < .01

β2

.1 ≤ β2 ≤ 5

2.8 < β2 ≤ 3.7

.1 ≤ β2 ≤ 2.8

δ3

.001 ≤ δ3 ≤ .02

.0056 < δ3 < .02

.001 ≤ δ3 ≤ .0056

δ5

.0005 ≤ δ5 ≤ .002

.0009 < δ5 ≤ .0011

.0005 ≤ δ5 ≤ .0009

12. Acknowledgements

634

We acknowledge P. Das for revision of the mathematical procedures and P. Merhotra for helpful discussion and writing assistance.

635

13. Author Contributions

633

638

Conceived and designed the ODE model: GP, SC. Performed the mathematical procedures: GP, SC. Performed the computational experiments: GP. Contributed to the writing of the manuscript: GP, KVSR, SC.

639

14. Bibliography

640

Neyrolles O, Mycobacteria and the Greasy Macrophage: Getting Fat and Frustrated. Infection and Immunity (2014); 82(2):472-475. doi: 10.1128/IAI.01512-13

636 637

641 642 643 644

Blaser MJ, The equilibria that allow bacterial persistence in human hosts. Nature, (2007); 449: doi:10.1038/nature06198

645

23

843-849.

646 647 648

Vergne I, Mycobacterium tuberculosis Phagosome Maturation Arrest: Mycobacterial Phosphatidylinositol Analog Phosphatidylinositol Mannoside Stimulates Early Endosomal Fusion. Molecular Biology of the Cell, (2004); 15(2): 751-760. doi: 10.1091/mbc.E03-05-0307

649 650

WHO, Tuberculosis Fact sheet 104. World Health Organization, (2010):1.

651 652 653

Miranda MS, The Tuberculous Granuloma: An Unsuccessful Host Defence Mechanism Providing a Safety Shelter for the Bacteria? Clinical and Developmental Immunology, (2012); 2012:14 doi: 10.1155/2012/139127

654 655 656

Suat LG, Protection of Mycobacterium tuberculosis from Reactive Oxygen Species Conferred by the mel2 Locus Impacts Persistence and Dissemination. Infection and Immunity, (2009); 77(6): 2557-2567. doi: 10.1128/IAI.01481-08

657 658 659

Mills CD, M1 and M2 Macrophages: Oracles of Health and Disease. Critical Reviews in Immunology, (2012); 32(6):463-488. doi: 10.1615/CritRevImmunol.v32.i6.10

660 661 662

Singh V, Mycobacterium tuberculosis-driven targeted recalibration of macrophage lipid homeostasis promotes the foamy phenotype. Cell Host and Microbes, (2012); 12(5):669-681. doi: 10.1016/j.chom.2012.09.012.

663 664 665

Mehrotra P, Pathogenicity of Mycobacterium tuberculosis Is Expressed by Regulating Metabolic Thresholds of the Host Macrophage. PLoS Pathogen, (2014); 10(7): e1004265. doi:10.1371/journal.ppat.1004265.

666 667 668

Vijay S, Asymmetric cell division in Mycobacterium tuberculosis and its unique features. Archives of Microbiology, (2014); 196(3):157-168 doi:10.1007/s00203-014-0953-7

669 670 671

Gil W, A replication clock for Mycobacterium tuberculosis. Nature Medicine, (2009); doi:10.1038/nm.1915.

15(2):

211-214.

672 673 674

Herbst S, Interferon Gamma Activated Macrophages Kill Mycobacteria by Nitric Oxide Induced Apoptosis. PLoS ONE, (2011); 6(5):e19105. doi:10.1371/journal.pone.0019105

675 676 677

Connelly L, Biphasic Regulation of NF-kB Activity Underlies the Pro- and Anti-Inflammatory Actions of Nitric Oxide. Journal of Immunology, (2001); 166:3873-3881. doi: 10.4049/

678 679 680 681

Nicholas A, Multi-Scale Modeling Predicts a Balance of Tumor Necrosis Factor-a and Interleukin-10 Controls the Granuloma Environment during Mycobacterium tuberculosis Infection, PLOS One, (2013); 8(7):e68680. doi:10.1371/journal.pone.0068680.

682 683 684

Corna G,Polarization dictates iron handling by inflammatory and alternatively activated macrophages. Haematologica, (2010); 95(11):1814-1822. doi: 10.3324/haematol.2010.023879

685 686 687

Kugelberg E, Immune evasion: Mycobacteria hide from TLRs. Nature Reviews Immunology, (2014); 14:6263. doi:10.1038/nri3604

688 689 690 691

Stephen W, Cytokine Responses During Mycobacterial and Schistosomal Antigen-Induced Pulmonary Granuloma Formation Production of Th 1 and Th2 Cytokines and Relative Contribution of Tumor Necrosis. American Journal of Pathology, (1994); 145(5):1105-1113. PMCID: PMC1887419

692 693 694

Hill H, M-1/M-2 Macrophages and the Th1/Th2 Paradigm. Journal of Immunology, (2000); 164:6166-6173. doi: 0022-1767/00/02.00

695 696 697

Weiss G, The Autocrine formation of hepcidin induces iron retention in human monocytes. Blood, (2008);111(4):23922399. doi:http://dx.doi.org/10.1182/blood-2007-05-090019

698 699 700

Doherty CP, Host-Pathogen Interactions: The Role of iron. Journal of Nutrition, (2007); 137(5): 1341-1344. doi: 17449603

701 702 703

Appelberg R, Macrophage nutriprive antimicrobial mechanism. Journal of Leukocyte Biology, (2006); 79(6):1117-1128. doi: 10.1189/jlb.0206079

704

24

705

Kim S, Role of nitric oxide in cellular iron metabolism. Biometals, (2003); 16(1):125-135.

706 707 708

Weiss G, Iron regulates nitric oxide synthase activity by controlling nuclear transcription. Journal of Experimental Medicine, (1994); 180(3): 969-976. doi:PMC2191642

709 710 711

Nairz M, Nitric oxide-mediated regulation of ferroportin-1 controls macrophage iron homeostasis and immune function in Salmonella infection. Journal of Experimental Medicine, (2013); 210(5):855-73. doi: 10.1084/jem.20121946

712 713 714

Suat L, Protection of Mycobacterium tuberculosis from Reactive Oxygen Species Conferred by the mel2 Locus Impacts Persistence and Dissemination. Infection and Immunity, (2009); 77(6): 2557-2567. doi: 10.1128/IAI.01481-08.

715 716 717

Kraml PJ, Iron loading increases cholesterol accumulation and macrophage scavenger receptor I expression in THP-1 mononuclear phagocytes. Metabolism, (2005); 54(4): 453459. doi: 10.1016/j.metabol.2004.10.012

718 719 720

Miner M, Role of cholesterol in Mycobacterium tuberculosis infection. Indian Journal of Experimental Biology, (2009); 47:407-411. PMID: 19634704

721 722 723

Gatfield J, Essential Role for Cholesterol in Entry of Mycobacteria into Macrophages. Science, (2000); 288(5471):16471651. doi: 10.1126/science.288.5471.1647

724 725 726

Ray JCJ, The timing of TNF and IFN-g signaling affects macrophage activation strategies during Mycobacterium tuberculosis infection. Journal of Theoretical Biology, (2008), 252(1): 2438. doi:10.1016/j.jtbi.2008.01.010

727 728 729

Lin Ling P, Modeling pathogen and host: in vitro, in vivo and in silico models of latent Mycobacterium tuberculosis infection. Drug Discovery Today, (2005); 2(2):149-154. doi: 10.1016/j.ddmod.2005.05.019

730 731 732

Arkhipov S, In vitro study of phenotypical characteristics of BCG granuloma macrophages over the course of granuloma development. Bullettin of Experimental Biology and Medicine, (2013);155(5):655-658. doi:0007-4888/13/1555-0655

733 734 735

Boelaert J,The Effect of the Host’s Iron Status on Tuberculosis. The Journal of Infectious Diseases, (2007); 195(12): 1745-1753. doi: 10.1086/518040

736 737 738 739

Chan ED, Induction of Inducible Nitric Oxide Synthase-NOz by Lipoarabinomannan of Mycobacterium tuberculosis Is Mediated by MEK1-ERK, MKK7-JNK, and NF-kB Signaling Pathways. Infection and Immunity, (2001); 69(4):2001-10. doi: 10.1128/IAI.69.4.2001-2010.2001

740 741 742

Chifmana J, The core control system of intracellular iron homeostasis: A mathematical model. Journal of Theoretical Biology, (2012); 300:9199. doi: 10.1016/j.jtbi.2012.01.024

743 744 745

Huang A, Lipid hydroperoxides inhibit nitric oxide production in RAW264.7 macrophages. Free Radical Biology and Medicine, (1999); 26(56):526537. doi: 10.1016/S0891-5849(98)00236-6

746 747 748

Yang X, Inhibition of inducible nitric oxide synthase in macrophages by oxidized low-density lipoproteins. Circulation Research, (1994); 74(2):318-28. doi: 10.1161/01.RES.74.2.318

749 750 751

Fuhrmana B, Iron induces lipid peroxidation in cultured macrophages, increases their ability to oxidatively modify LDL, and affects their secretory properties. Atherosclerosis, (1994); 111(1):6578. doi: 10.1016/0021-9150(94)90192-9

752 753 754

Recalcati S, Differential regulation of iron homeostasis during human macrophage polarized activation. European Journal of Immunology, (2010); 40:824835. doi: 10.1002/eji.200939889

755 756 757

Serafin-Lopez J, The Effect of Iron on the Expression of Cytokines in Macrophages Infected with Mycobacterium tuberculosis. Scandinavian Journal of Immunology, (2004); 60:329337. doi: 10.1111/j.0300-9475.2004.01482.x

758 759 760

Kirschner D, Mathematical and computational approaches can complement experimental studies of hostpathogen interactions. Cellular Microbiology, (2009); 11: 531539 doi:10.1111/j.1462-5822.2008.01281.x

761 762 763

Wigginton J, A model to predict cell-mediated immune regulatory mechanisms during human infection with Mycobacterium tuberculosis. Journal of Immunology, (2001) 166:19511967. doi: 10.4049/jimmunol.166.3.1951

25

764 765 766

Stewart G, Tuberculosis: doi:10.1038/nrmicro749

a problem with persistence. Nature Reviews Microbiology, (2003);

1:97-105.

767 768 769

Antia R, Models of the within-host dynamics of persistent mycobacterial infections. Proceedings Of The Royal Society Of London, (1996); 263(1368): 257-263. doi; 009032696

770 771 772

Magombedze G, A mathematical representation of the development of Mycobacterium tuberculosis active, latent and dormant stages. Journal of Theoretical Biology, (2012); 292:44-59. doi: 10.1016/j.jtbi.2011.09.025

773 774 775

Zhilan F, A Model for Tuberculosis with Exogenous Reinfection. Theoretical Population Biology, (2000); 57:235-247. doi:10.1006-tpbi.2000.1451

776 777 778

Herbert W, The Mathematics of doi:10.1137/S0036144500371907

Infectious

Diseases.

SIAM

Review,

(2000),

42(4):

599-653.

779 780 781

Das B, CD271(+) bone marrow mesenchymal stem cells may provide a niche for dormant Mycobacterium tuberculosis. Science Translational Medicine, (2013); 5(170):170ra13 doi: 10.1126/scitranslmed.3004912:

782 783 784

Pienaara E, A mathematical model of the initial interaction between Mycobacterium tuberculosis and macrophages. Journal of Theoretical Biology, (2014), 342:2332. doi: 10.1016/j.jtbi.2013.09.029

785 786 787

Wayne LG, Dormancy of Mycobacterium tuberculosis and latency of disease. European Journal of Clinical Microbiology and Infectious Diseases, (1994);13(11):908-14.doi: 10.1007/BF02111491

788 789 790

Ford C, Use of whole genome sequencing to estimate the mutation rate of Mycobacterium tuberculosis during latent infection. Nature Genetics, (2011); 43:482486 doi:10.1038/ng.811

791 792 793

Saini D, Ultra-low dose of Mycobacterium tuberculosis aerosol creates partial infection in mice. Tuberculosis, (2012); 92(2):160-5. doi: 10.1016/j.tube.2011.11.007

794 795 796

Nagumo N, Uber die lage der integralkurven gewnlicherdifferantialgleichungen, Proc. Phys. Math. Soc. Jpn., (1942); 24:551.

797 798 799

Wei-Min L, Criterion of Hopf Bifurcations Without Using Eigenvalues. Journal of Mathematical analysis and applications, (1994); 182(1):250-256. doi: 10.1006/jmaa.1994.1079

800 801 802

Rees E, Graphical Discussion of the Roots of a Quartic Equation. The American Mathematical Monthly, (1992); 29 (2):51-55. DOI: 10.2307/2972804

803 804 805

Hurwitz A, ”Uber die Bedingungen unter welchen eine Gleichung nur Wurzeln mit negativen reellen Teilen besitzt”. Math. Ann., (1895) 46:273-284.

806 807 808

Dannenberg AM, Macrophage turnover, division and activation with developing, peak and ”healed” tuberculous lesions produced in rabbits by BCG. Tuberculosis, (2003); 83(4):251-60. doi: 10.1016/S1472-9792(03)00048-9

809 810 811

Sibley LD, Mycobacterial lipoarabinomannan inhibits gamma interferon-mediated activation of macrophages. Infection and Immunity, (1988), 56(5); 1232-1236. PMC259795

812 813 814

Denis M, Interferon-gamma-treated murine macrophages inhibit growth of tubercle bacilli via the generation of reactive nitrogen intermediates. Cellular Immunology, (1991); 132(1):150-157. PMID: 1905984

815 816 817 818

Flesch I, Mycobacterial growth inhibition by interferon-gamma-activated bone marrow macrophages and differential susceptibility among strains of Mycobacterium tuberculosis. The Journal of Immunology, (1987); 138(12): 44084413. doi: 0022-1767/87/13812-4408 02.00/0

819 820

Bogdan C, Nitric oxide and the immune response. Nature Immunology, (2001); 2(10): 907-916. doi:10.1038/ni1001-907

821 822

Umansky V, Co-stimulatory effect of nitric oxide on endothelial NF-B implies a physiological self-amplifying

26

823 824

mechanism. European Journal of Immunology, (1998); 4141(199808)28:08¡2276::AID-IMMU2276¿3.0.CO;2-H

28:22762282. doi:

DOI: 10.1002/(SICI)1521-

825 826 827

Bhat KH, PPE2 protein of Mycobacterium tuberculosis may inhibit nitric oxide in activated macrophages. Ann NY Academy of Science, (2013);1283:97-101. doi: 10.1111/nyas.12070

828 829 830

Collins H, Withholding iron as a cellular defence mechanism-friend or foe? (2008);38(7):18031806. doi: 10.1002/eji.200838505

European Journla of Immunology,

831 832 833

Isanaka S, Iron deficiency and anemia predict mortality in patients with tuberculosis. Journal of Nutrition, (2012);142(2):350-357. doi: 10.3945/jn.111.144287

834 835

Weiss G, Iron and anemia of chronic disease. Kidney International, (1999); 55:1217; doi:10.1046

836 837 838

Nemeth J, IL-6 mediates hypoferremia of inflammation by inducing the synthesis of the iron regulatory hormone hepcidin. Clin Invest. (2004); 113(9):12711276. doi:10.1172/JCI20945

839 840 841

Tufariello J, Latent tuberculosis: mechanisms of host and bacillus that contribute to persistent infection. THE LANCET Infectious Diseases (2003) 3(9): 578590. doi:10.1016/S1473-3099(03)00741-2.

842 843 844

Marino S, A Methodology For Performing Global Uncertainty And Sensitivity Analysis In Systems Biology J Theor Biol., (2009); 254(1): 178196. doi:10.1016/j.jtbi.2008.04.011.

845 846 847

Mckay MD, Mckay MD, Beckman RJ, Conover WJ. Comparison of 3 Methods for Selecting Values of Input Variables in the Analysis of Output from a Computer Code. Technometrics. 1979;21(2):239245.

848

27

Mathematical model of mycobacterium-host interaction describes physiology of persistence.

Despite extensive studies on the interactions between Mycobacterium tuberculosis (M.tb) and macrophages, the mechanism by which pathogen evades anti-m...
2MB Sizes 0 Downloads 8 Views