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Soft Matter Accepted Manuscript
This article can be cited before page numbers have been issued, to do this please use: S. Najem and M. Grant, Soft Matter, 2014, DOI: 10.1039/C4SM01842G.
This is an Accepted Manuscript, which has been through the Royal Society of Chemistry peer review process and has been accepted for publication. Accepted Manuscripts are published online shortly after acceptance, before technical editing, formatting and proof reading. Using this free service, authors can make their results available to the community, in citable form, before we publish the edited article. We will replace this Accepted Manuscript with the edited and formatted Advance Article as soon as it is available. You can find more information about Accepted Manuscripts in the Information for Authors. Please note that technical editing may introduce minor changes to the text and/or graphics, which may alter content. The journal’s standard Terms & Conditions and the Ethical guidelines still apply. In no event shall the Royal Society of Chemistry be held responsible for any errors or omissions in this Accepted Manuscript or any consequences arising from the use of any information it contains.
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DOI: 10.1039/C4SM01842G
The Chaser And The Chased: A Phase-Field Model Of An Immune Response
In this paper we present a model for an immune response to an invading pathogen. Particularly, we follow the motion of a neutrophil as it migrates to the site of infection guided by chemical cues, a mechanism termed chemotaxis, with the ability to reorient itself as the pathogen changes its position. In the process, the cell undergoes morphological alterations, in addition to the structural changes observed at its leading edge. Also, we derive a condition for a successful immune reaction by relating the speed of the neutrophil to that of the pathogen and to the diffusion coefficient of the chemical attractant.
1
Introduction
The response of the immune cells to pathogens such as viruses, bacteria, microbes, and cancerous cells, and the ability to distinguish them from healthy tissues are critical to the longevity of any living system. The detection of those triggers a series of reactions that releases chemicals at the site of infection or inflammation in order to recruit immune cells 1–4 . For example, among the first responders to the invading pathogens are the neutrophils. These, abundant in the blood stream, constitute the bulk of the leucocytes cells making up around 60% of their total number 5 . They migrate to the site of infection as a response to a chemical signal, a process termed chemotaxis, and undergo shape deformation throughout this motion. Subsequent to the activation of a neutrophil, proteins and lipids accumulate at the cell’s leading edge facing the highest chemical concentration and forming what is called the plasma membrane 6 . There are two types of cell movement: the first involves single cell migration, which is instrumental in understanding physiological motility mechanisms. It enables cells to arrange themselves in tissues, which is the case in morphogenesis or to “transiently pass through the tissue, as shown by immune cells” 7 . The other, collective motion, involves the continuous connectedness of cells as they move; examples include embryogenesis, wound healing, and cancer metastasis 8 . Leucocytes, and amongst them neutrophils, move according to the single-cell mode and act independently 9,10 and this aspect is what we seek to recover in our work. Particularly, modeling their motion is challenging as they experience membrane alterations and an emergence of a leading edge; both of which make their morphodyamics quite intricate. Studies have used level-set 11,12 and phase-field method in order to follow their morphology 13 , however they have failed to capture these coma
Department of Ecology and Evolutionary Biology, Princeton University, Princeton NJ, USA. E-mail: [email protected] b Physics Department Rutherford Building McGill University, 3600 rue University, Montr´eal, Qu´ebec, Canada
plex boundary deformations and neglected the presence of the plasma membrane. A more recent model 14 recovered these dynamics in a simplified setting where the neutrophil moves to a fixed site. In this paper, we build a phase-field model in which we reproduce most of these dynamical processes and include the ability of the cell to reorient itself by coupling its motion to that of a bacteria invading the system and trying to escape the neutralization 8 ; it evades phagocytic cells by detecting peroxide, hypochlorite, and N-chlorotaurine, which result from the respiratory bursts of immune cells. In addition to that, we predict a critical speed of the neutrophil below which the immune response fails to rid the system of the pathogens, which is pivotal to the progression or the termination of any disease. This particular level of description in understanding the workings of the “two-body problem”, that of an immune responder and a pathogen, is a bridge to form the bigger picture involving the coordination between the different immune respondents. 1.1
Theoretical Background
The microscopic level interactions between particles explain the existence of phases of matter. These differences between phases can be reflected at the macroscopic one and are represented by a phase-field or an order parameter φ . A simple example would be the Ising model where φ quantifies the total magnetization 15 . Regions where φ = 0 designate the disordered phase, whereas regions where φ = 1 distinguish the ordered one. This spatially varying order parameter φ (r) could be driven to order by applying an external magnetic field H. Now including all the above mentioned details and in order to describe the system, particularly the dynamics of φ , an energy functional F is constructed assuming it is analytic. This is the Ginzburg-Landau premise that enables the construction of F as a series expansion in powers of the order parameter φ and its derivatives by conserving the symmetries of the system 16 . Then a variational derivative is applied to this functional to determine the dynamics of φ depending on the nature 1–6 | 1
Soft Matter Accepted Manuscript
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Sara Najem,∗a and Martin Grantb
Soft Matter
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of this evolving field (being conserved or not). The most general form of the functional includes a free-energy of the bulk: a double-well which is the expansion in powers of φ ; its minima are the values assigned to the coexisting phases. Furthermore, odd powers of ∇φ have zero coefficients and are dropped out of the equation of F in order to satisfy translational invariance given by x → −x. The even powers of ∇φ are cut to second order to include the interaction between phases and the presence of an interface. F in the presence of an external field H becomes: F=
Z
ε(∇φ )2 + f (φ ) + Hφ ]dV, [ 2
Model Assumptions
The neutrophil can be clearly macroscopically distinguished from its surrounding. Therefore, these can be thought of as two different coexisting phases. Then the field φ1 is assigned the values φ1 = 1 and φ1 = 0 to distinguish the cell and its environment respectively. Similarly, the pathogens is depicted by φ2 , which is 1 inside the bacteria and zero elsewhere. Additionally, we describe the emerging plasma membrane by ψ = 1 inside that region and 0 elsewhere. The neutrophil secretes a chemical denotes c1 that repels the pathogens 19 . On the other hand, the bacteria is tagged by a chemical c2 , which acts as an attractant to the neutrophil. Therefore, both the velocities of the chaser V1 and the chased V2 are assumed to depend on the gradients of the chemical attractant and repellent respectively 20 , with the appropriate choices of signs, as illustrated in Fig. 1.
2
Equations
Now in order to describe the system fully Eq. 1 is modified to account for the presence of these different materials φ1 , φ2 , and ψ. Therefore, to guarantee the coexistence of their 2|
1–6
2
2
2) for the plasma membrane, and ε3 (∇φ for the trophil, ε2 (∇ψ) 2 2 bacteria. Similarly, εi are proportional to surface tension. The external fields driving both the neutrophil and pathogens are denoted by H1 and H2 . Now the functional is given by:
F=
(1)
where f (φ ) = aφ 2 (1 − φ )2 , where a has units of energy per volume, and ε is proportional to surface tension. The doublewell f (φ ) is minimized at φ = 1 and φ = 0. In this case, φ denotes the magnetization; its dynamics are given by this nonconservative partial differential equation, which in the presence of stochastic terms is commonly known as Model A: ∂φ δF 16,17 . On ∂t = −Γ δ φ , where Γ is the mobility of the field the other hand, if φ was a conserved quantity such as concentration, it should have followed a conservative dynamics or Model B given by: ∂∂tφ = Γ∇2 δδF φ . Kinetics terms that cannot be captured by the functional should be added to both types of equations. One such example, in which a kinetic term appears as a nonlinearity, is the Kardar-Parisi-Zhang (KPZ) equation 18 . 1.2
phases the double-well is modified as such: f (φ1 , φ2 , ψ) = aφ12 (1 − φ1 )2 + aφ22 (1 − φ2 )2 + aψ 2 (1 − ψ)2 . Also, the terms accounting for the interaction between phases of the same ma2 1) for the neuterial are added. These are given by: ε1 (∇φ 2
ε1 (∇φ1 )2 ε2 (∇ψ)2 ε3 (∇φ2 )2 + + + 2 2 2 f (φ1 , φ2 , ψ) + H1 φ1 + H2 φ2 ]dV.
Z
[
(2)
The neutrophil and the pathogen both conserve their volumes. Therefore, their dynamics are conservative. Explicitly, their ∂ φ1 2 δF 1 equations are given by dφ dt = γ1 ∇ δ φ = ∂t + V1 · ∇φ1 , and 1
∂ φ2 = γ3 ∇2 δδ F φ2 = ∂t +V2 · ∇φ2 , where γ1 and γ3 are the mobilities, V1 and V2 are the velocities of the neutrophil and the bacteria respectively. We define V1 = b1 ∇c2 and V2 = −b2 ∇c1 in order to guarantee the attraction of the neutrophil by the pathogen and the repulsion of the latter in the process. Here the terms b1 and b2 act as amplifiers to the sensitivity to chemical cues. These now allow us to write the following equations: dφ2 dt
(
∂ φ1 ∂t ∂ φ2 ∂t
= −γ1 ∇2 (ε1 ∇2 φ1 − fφ1 − H1 ) − b1 ∇c2 · ∇φ1 , = −γ3 ∇2 (ε3 ∇2 φ2 − fφ2 − H2 ) + b2 ∇c1 · ∇φ2 ,
(3)
where fφ1 and fφ2 are the derivatives of f with respect to φ1 and φ2 respectively. Now by choosing γ1 ∇2 H1 = −b1 c2 ∇φ1 and γ1 ∇2 H2 = b2 c1 ∇φ1 we ensure that both equations of φ1 and φ2 can be written as continuity equations. Consequently, we impose the conservation of their total volume. Now the dynamics of the leading edge is not conservative as it disappears once the neutrophil reaches the highest chemical ∂ψ δF concentration site 5 . Therefore, dψ dt = −γ2 δ ψ = ∂t +V · ∇ψ, where γ2 is the mobility of the leading edge and V its velocity. This region of the neutrophil moves in synchrony with the rest of the cell and is driven by the chemical attractant c2 . Therefore, we define V = b1 φ1 ∇c2 . The term φ1 appearing in the definition of V couples the motion of the plasma membrane to the rest of the cell. Fig. 1 illustrates the system. The neutrophil secretes c1 . Therefore, it is advected by the same velocity field V1 as φ1 . This in addition to the diffusion and the degradation effects. Similar argument holds for the equation of c2 , which is convected by the same velocity field V2 as the pathogen. The equations governing the dynamics of the system can now be derived by applying the variational derivative to F in Eq. 2 and the results of Eq. 3 to give:
Soft Matter Accepted Manuscript
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DOI: 10.1039/C4SM01842G
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Fig. 1 This figure shows the geometry of the system, reveals the spatial dependence of the fields, and gives a hint on how the velocities are defined. The neutrophil is depicted by φ = 1, the pathogen by φ2 = 1, and the leading edge is shown in red by ψ = 1. The choices of the dependence of the velocities on chemical gradient with proper sign is illustrated.
∂ φ1 2 2 ∂t = −γ1 ∇ (ε1 ∇ φ1 − fφ1 ) − b1 ∇(φ1 ∇c2 ) ∂ ψ ∂t = γ2 (ε2 ∇2 ψ − fψ ) − (b1 φ1 ∇c2 ) · ∇ψ ∂ c1 2 ∂t = D1 ∇ c1 − (b1 ∇c2 ) · ∇c1 − e1 c1 ∂ φ2 2 2 ∂t = −γ3 ∇ (ε3 ∇ φ2 − fφ2 ) + b2 ∇(φ2 ∇c1 ) ∂ c2 = D ∇2 c + (b ∇c ) · ∇c − e c 2 2 2 1 2 2 2 ∂t
(4)
where Di are the diffusion coefficient, ei are the degradation rates (for i = 1, 2 and 3), and fψ is the derivative of f with respect to ψ.
3
Fig. 2 The figure shows snapshots of the dynamics of all three fields φ1 , ψ and φ2 representing respectively the neutrophil, its polymerized region, and the chased cell. They are initially at the lower leftmost part of the domain, and are tracked as they move to its upper rightmost region. Along the way, the neutrophil’s morphological changes can be observed, portrayed by yellow to dark orange cell. Also the evolution of its plasma membrane at its leading edge is depicted by a red color all throughout. The chased cell’s field color ranges from light blue to light green.
Results
We solve the system of Eq. 4 in two dimensions as shown in Fig. 2. The values of γ1 , γ2 , as well as ε1 and ε2 are taken from 13 ; these are experimentally measured parameters. As for the other parameters, unique to our model, we explored a range of values and admissible ones satisfy a condition, which we will derive in the course of our discussion. The neutrophil, its leading edge, and the pathogens are shown at different times during the chase. We notice the amount of deformation the cell undergoes, in addition to the appearance of the leading edge at the left most part of the figure and its disappearance once the pathogen is reached at the rightmost part of the figure, which is consistent with observations. Now we calculate the mean displacement of the center of mass of the neutrophil as a function of time and compare our results with experimental cell motility assays 21,23
and the model given by Parkhurst and Saltzman 22 . It is given by the following equation: D2 (t) = 4µ(t − P + Pe−t/P ), where P is its persistence time and µ is the random motility coefficient. We model the directional motion of cells while 22 models random walks, however both follow the motion of cells in their diffusive states and hence their equivalence. The non-dimensionalised time will allow us to rewrite ′ D2 (t ′ ) = 4µP(t ′ − 1 + e−t ), where t ′ = t/P. Then the product µ × P is essential in determining the significance of our result. In Fig. 3(a) we show the square displacement defined as D2 (t) = (xcm (t) − xcm (t − 1))2 + (ycm (t) − ycm (t − 1))2 , where xcm and ycm are the coordinates of the neutrophil’s center of mass. D2 (t) will decrease as the cell approaches the pathogen. Therefore, we truncate it to exclude the regime where the neutrophil slows down, as shown in Fig. 3(b). The conversion from pixels to µm is done by choosing the diameter of the cell in pixels to correspond to d = 8.5±0.5µm, which is the average diameter of a neutrophil. The numerical scheme used in solving Eq. 4 has an error of the order of h2x , where the latter is the grid step size. The square displacement computed from the numerical solution of our model then has an error of the order of h4x . The equation of D2 , fitted to that extracted from the numerical simulation, is shown in red in Fig. 3 and predicts a persistence time of 1–6 | 3
Soft Matter Accepted Manuscript
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DOI: 10.1039/C4SM01842G
Soft Matter
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DOI: 10.1039/C4SM01842G
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Fig. 4 The figure shows the evolution of system’s components. The position of the neutrophil with respect to that of the pathogen and the level curve c2 of the repellant is followed.
(b)
Fig. 3 In (a) the square of the displacement of the neutrophil’s center of mass as a function of iteration number is shown, whereas in (b) the neutrophil is tracked before reaching the pathogen and the truncated displacement of (a) is plotted. The red curve is the fit of the model given by: D2 (t) = 4µ(t − P + Pe−t/P ). Also, the numerical scheme used to solving Eq. 4 and thus determining the position of the neutrophil has an error of the order of h2x , where hx is the grid step size. This defines the error bar on the square displacement to be h4x .
4|
1–6
Finally, we can make a prediction for the condition under which the neutrophil will fail to neutralize the pathogen and the immune response is then inefficient. We take the limit where the degradation terms are slow compared to the convective ones and thus set e1 and 2 to 0. Then c1 and c2 are now Gaussian advected by the flow and are centered at the cell’s and pathogen’s centers of mass respectively. Fig. 4 is an illustration of the dynamics. The new position of the center of mass of the pathogen after one iteration is given by dc1 1 (x p , y p ) = (x p0 − b2t dc dx , y p0 − b2 t dy ), where x p0 and y p0 are the initial pathogen’s center of mass coordinate, and t is the time step size. Similarly, the neutrophil in this time step has dc2 2 moved (xn , yn ) = (xn0 + b1t dc dx , yn0 + b1 t dy ), where xn0 and yn0 are the initial coordinates of the neutrophil’s closest point
Soft Matter Accepted Manuscript
P = 1 [iteration] and a motility µ = 19.2 × 10−8 [µm2 per iteration]. For a collagen concentration of 0.1mg/ml, for example, the motility is µexperimental = 0.51 ± 0.04µm2 /s and Pexperimental = 40 ± 25s 22 . Their product A = 20 ± 15[µm2 ]. This interval is in agreement with our numerical results, where µ × P = 19 ± 1[µm2 ]. For different collagen concentrations the value of P is still consistent with the experimentally measured values. This shows that the model is capable of reproducing physically measurable properties of the system and confirms the validity of the assumptions made about the velocity dependence on the chemical gradient.
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√ to the pathogen. c2 , which is a Gaussian of width w = 4D2t, defines a level curve in which the neutrophil must partially always lie. This is translated by the following inequality: (5)
which should hold for all times. This condition can be put to experimental test to check its validity and confirm our findings. In 1-D, Eq. 5 then becomes the difference between the neudc2 1 tropil and pathogen’s velocities: b2 dc dx + b1 dx = vn − v p > √ x p0 −xn0 −2 D2 t . t
In Fig. 5 we show, for the 1D motion, the region of parameter space where the immune response is successful. In 2D the condition is defined in a 4 dimensional parameter space.
2
1.5
1
Fig. 5 The plot shows the condition for a successful immune response relating the difference in the neutrophil’s and pathogen’s velocities versus the difference in their initial positions. In the area above the curve the immune response is effective.
Neutrophil s Decision
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(x p − xn )2 + (y p − yn )2 < w2 ,
Further, the neutrophil is affected by intermediary chemoattractants such as IL−8 and LTB4 , which are encountered during its migration to the site of infection 27 . Therefore, the need to prioritize in the presence of competing signals becomes vital. We establish a dose-response curve in the case of two conflicting chemoattractants by varying the distance between them as well as their corresponding cues’ sensitivities, or velocity amplifications b11 and b12 . We follow the decision of the neutrophil and assign the value 1 in case it migrates to site 1 and −1 otherwise. Fig. 6 suggests that the neutrophil is distracted by intermediary cues; that is when it chooses to migrate to site 2. This is in total agreement with experimental observations of Pten−/− neutrophils 28 . However, the neutrophil can selectively ignore these cues using PTEN-dependent mechanism regardless of their concentrations 28 . Therefore, at each time step we rescale our parameter b11 in an adaptive way to achieve directional migration to site 1 inhibiting the effect of other cues and replicating the effect of the PTEN mechanism while otherwise fixing b11 would recover the distracting effect of intermediary cues in Pten−/− neutrophils.
0.5
0
−0.5
−1
−1.5
Therefore, modifying b1 , which is the velocity amplification factor of the neutrophil, could have drastic effects on the dynamics. Explicitly, the effect of chemotherapy on breast cancer is known to lead to a failure in neutrophil chemotactic function, specifically to stimuli like fMLP and LTB4 26 . In other words, a pathogen would escape neutralization in sick individuals undergoing chemotherapy while in healthy individuals that same pathogen would be successfully neutralized. This reduced migratory capacity of the neutrophils after chemotherapy increases the risk of infection, which our model predicts when b1 is reduced in accordance with reported experimental findings 26 , which establishes a relation between the parameter b1 and the absence of immunosuppressive factors.
−2 6 5
Soft Matter Accepted Manuscript
DOI: 10.1039/C4SM01842G
6
4
5 3
4 3
2 2 1
Distance Between Conflicting Signals
1 0
0
Ratio of Sensitivities to Chemical Attractants
Fig. 6 The plot shows the decision of the neutrophil to go to either site 1 or site 2 as a function of the chemical cues’ sensitivities and the distance between the sites. 1 corresponds to migrating to site 1 and −1 to site 2.
1–6 | 5
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DOI: 10.1039/C4SM01842G
Conclusions
In this paper, we have used the phase-field method to replicate an immune response. We have reproduced the membrane deformations the neutrophil undergoes as it responds to chemical cues. In addition, we have derived an inequality that relates the speed of the neutrophil to that of the pathogen and the diffusion coefficient of the chemical attractant that the latter secretes. This is a consequential result that models how the immune system succeeds in fighting the pathogens or fails based on three major parameters, the pathogens and the neutrophil’s sensitivities to chemical cues, and the diffusion coefficient of the attractant. Tuning these parameters is key to a successful immune reaction. Additionally, in contrast to experimental setups, which under agarose neutrophil migration assays are either unidirectional or establish a limited number of chemical fields, we can generate complex chemotactic fields and develop numerical single cell assays that would help in pharmaceutical treatments. We have presented an additional application to phase-field in the context of active matter 25 and proved its efficiency in reproducing salient features of the dynamical processes at play.
5
Acknowledgments
We acknowledge support from the Natural Sciences and Engineering Research Council of Canada and le Fonds de recherche du Qu´ebec — Nature et technologies.
References 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
6|
P.A. Iglesias and P.N. Devreotes, Curr. Opin. Cell Biol., 2008, 20, 35. C.A. Parent and P.N. Devreotes, Science, 1999 284, 765. G. William et al., J. Leukocyte Biol., 2006, 79. L.V. Dekker and A.W Segal, Science, 2000, 287, 982. M. Eisenbach. Chemotaxis (Imperial College Press, 2004). A. Gamba et al., Proc. Natl. Acad. Sci., 2005, 102, 16927 P. Friedl and D. Gilmour. Nature Reviews Molecular cell biology, 2009, 10, 445. L. Benov and I. Fridovich. Proc. Natl. Acad. Sci., 1996, 93, 4999. B. Alberts et al., Molecular Biology of the Cell (Garland Science 2002). N.S Gov. HFSP Journal,2009, 3. Yangi et al., BMC Syst. Biol.,2008, 2, 68. C.W. Wolgemuth and M. Zajac, J. Comput. Phys., 2010, 229, 7287. D. Shao, H. Levine, and W.J. Rappel, Proc. Natl. Acad. Sci., 2012, 109, 6851. S. Najem and M. Grant. Phys. Rev. E , 2013, 88, 034702. D. Ruelle, Statistical Mechanics: Rigorous Results (New York: W.A. Benjamin Inc,1969) N. Provatas and K. Elder, Phase Field Methods in Materials Science and Engineering (WILEY-VCH, 2010). Chaikin P.M, Lubensky T.C. Principles of Condensed Matter (Cambridge University Press, 1995) M. Kardar, G. Parisi, and Y.C Zhang, Phys. Rev. Lett., 1986, 56, 889. J.E Gestwicki and L.L Kiessling, Nature, 2002, 415, 81.
1–6
20 E.F Keller and L.A Segel, J. Theoret. Biol, 1971, 30, 225. 21 P.V Moghea, R.D Nelsonb, and R.T Tranquillo, J Immunol. Methods, 1995, 180, 193. 22 M.R Parkhurst and W.M Saltzman, Biophys. J., 1992, 61, 306. 23 F. Lin et al., Biochem. Bioph. Res. Co., 2004, 319, 576. 24 S.H Zigmond, Nature, 1974, 249, 450. 25 S. Najem and M. Grant, Europhys. Lett., 2013, 102, 16001. 26 M.A, Mendona et al, Cancer chemotherapy and pharmacology, 2006, 57 (5), 663. 27 B. Heit, Bryan et al.,The Journal of cell biology, 2002, 159, 91. 28 B Heit et al., Nature immunology, 2008, 9, 743.
Soft Matter Accepted Manuscript
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4
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DOI: 10.1039/C4SM01842G
We model the interplay between a neutrophil and a pathogen, follow their morphodynamics, and identify a condition for the success of the immune response.
Soft Matter Accepted Manuscript
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TABLE OF CONTENTS
1
Soft Matter Accepted Manuscript
This article can be cited before page numbers have been issued, to do this please use: S. Najem and M. Grant, Soft Matter, 2014, DOI: 10.1039/C4SM01842G.
This is an Accepted Manuscript, which has been through the Royal Society of Chemistry peer review process and has been accepted for publication. Accepted Manuscripts are published online shortly after acceptance, before technical editing, formatting and proof reading. Using this free service, authors can make their results available to the community, in citable form, before we publish the edited article. We will replace this Accepted Manuscript with the edited and formatted Advance Article as soon as it is available. You can find more information about Accepted Manuscripts in the Information for Authors. Please note that technical editing may introduce minor changes to the text and/or graphics, which may alter content. The journal’s standard Terms & Conditions and the Ethical guidelines still apply. In no event shall the Royal Society of Chemistry be held responsible for any errors or omissions in this Accepted Manuscript or any consequences arising from the use of any information it contains.
www.rsc.org/softmatter
Page 1 of 7
Soft Matter View Article Online
DOI: 10.1039/C4SM01842G
The Chaser And The Chased: A Phase-Field Model Of An Immune Response
In this paper we present a model for an immune response to an invading pathogen. Particularly, we follow the motion of a neutrophil as it migrates to the site of infection guided by chemical cues, a mechanism termed chemotaxis, with the ability to reorient itself as the pathogen changes its position. In the process, the cell undergoes morphological alterations, in addition to the structural changes observed at its leading edge. Also, we derive a condition for a successful immune reaction by relating the speed of the neutrophil to that of the pathogen and to the diffusion coefficient of the chemical attractant.
1
Introduction
The response of the immune cells to pathogens such as viruses, bacteria, microbes, and cancerous cells, and the ability to distinguish them from healthy tissues are critical to the longevity of any living system. The detection of those triggers a series of reactions that releases chemicals at the site of infection or inflammation in order to recruit immune cells 1–4 . For example, among the first responders to the invading pathogens are the neutrophils. These, abundant in the blood stream, constitute the bulk of the leucocytes cells making up around 60% of their total number 5 . They migrate to the site of infection as a response to a chemical signal, a process termed chemotaxis, and undergo shape deformation throughout this motion. Subsequent to the activation of a neutrophil, proteins and lipids accumulate at the cell’s leading edge facing the highest chemical concentration and forming what is called the plasma membrane 6 . There are two types of cell movement: the first involves single cell migration, which is instrumental in understanding physiological motility mechanisms. It enables cells to arrange themselves in tissues, which is the case in morphogenesis or to “transiently pass through the tissue, as shown by immune cells” 7 . The other, collective motion, involves the continuous connectedness of cells as they move; examples include embryogenesis, wound healing, and cancer metastasis 8 . Leucocytes, and amongst them neutrophils, move according to the single-cell mode and act independently 9,10 and this aspect is what we seek to recover in our work. Particularly, modeling their motion is challenging as they experience membrane alterations and an emergence of a leading edge; both of which make their morphodyamics quite intricate. Studies have used level-set 11,12 and phase-field method in order to follow their morphology 13 , however they have failed to capture these coma
Department of Ecology and Evolutionary Biology, Princeton University, Princeton NJ, USA. E-mail: [email protected] b Physics Department Rutherford Building McGill University, 3600 rue University, Montr´eal, Qu´ebec, Canada
plex boundary deformations and neglected the presence of the plasma membrane. A more recent model 14 recovered these dynamics in a simplified setting where the neutrophil moves to a fixed site. In this paper, we build a phase-field model in which we reproduce most of these dynamical processes and include the ability of the cell to reorient itself by coupling its motion to that of a bacteria invading the system and trying to escape the neutralization 8 ; it evades phagocytic cells by detecting peroxide, hypochlorite, and N-chlorotaurine, which result from the respiratory bursts of immune cells. In addition to that, we predict a critical speed of the neutrophil below which the immune response fails to rid the system of the pathogens, which is pivotal to the progression or the termination of any disease. This particular level of description in understanding the workings of the “two-body problem”, that of an immune responder and a pathogen, is a bridge to form the bigger picture involving the coordination between the different immune respondents. 1.1
Theoretical Background
The microscopic level interactions between particles explain the existence of phases of matter. These differences between phases can be reflected at the macroscopic one and are represented by a phase-field or an order parameter φ . A simple example would be the Ising model where φ quantifies the total magnetization 15 . Regions where φ = 0 designate the disordered phase, whereas regions where φ = 1 distinguish the ordered one. This spatially varying order parameter φ (r) could be driven to order by applying an external magnetic field H. Now including all the above mentioned details and in order to describe the system, particularly the dynamics of φ , an energy functional F is constructed assuming it is analytic. This is the Ginzburg-Landau premise that enables the construction of F as a series expansion in powers of the order parameter φ and its derivatives by conserving the symmetries of the system 16 . Then a variational derivative is applied to this functional to determine the dynamics of φ depending on the nature 1–6 | 1
Soft Matter Accepted Manuscript
Published on 29 October 2014. Downloaded by Brown University on 31/10/2014 00:42:53.
Sara Najem,∗a and Martin Grantb
Soft Matter
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of this evolving field (being conserved or not). The most general form of the functional includes a free-energy of the bulk: a double-well which is the expansion in powers of φ ; its minima are the values assigned to the coexisting phases. Furthermore, odd powers of ∇φ have zero coefficients and are dropped out of the equation of F in order to satisfy translational invariance given by x → −x. The even powers of ∇φ are cut to second order to include the interaction between phases and the presence of an interface. F in the presence of an external field H becomes: F=
Z
ε(∇φ )2 + f (φ ) + Hφ ]dV, [ 2
Model Assumptions
The neutrophil can be clearly macroscopically distinguished from its surrounding. Therefore, these can be thought of as two different coexisting phases. Then the field φ1 is assigned the values φ1 = 1 and φ1 = 0 to distinguish the cell and its environment respectively. Similarly, the pathogens is depicted by φ2 , which is 1 inside the bacteria and zero elsewhere. Additionally, we describe the emerging plasma membrane by ψ = 1 inside that region and 0 elsewhere. The neutrophil secretes a chemical denotes c1 that repels the pathogens 19 . On the other hand, the bacteria is tagged by a chemical c2 , which acts as an attractant to the neutrophil. Therefore, both the velocities of the chaser V1 and the chased V2 are assumed to depend on the gradients of the chemical attractant and repellent respectively 20 , with the appropriate choices of signs, as illustrated in Fig. 1.
2
Equations
Now in order to describe the system fully Eq. 1 is modified to account for the presence of these different materials φ1 , φ2 , and ψ. Therefore, to guarantee the coexistence of their 2|
1–6
2
2
2) for the plasma membrane, and ε3 (∇φ for the trophil, ε2 (∇ψ) 2 2 bacteria. Similarly, εi are proportional to surface tension. The external fields driving both the neutrophil and pathogens are denoted by H1 and H2 . Now the functional is given by:
F=
(1)
where f (φ ) = aφ 2 (1 − φ )2 , where a has units of energy per volume, and ε is proportional to surface tension. The doublewell f (φ ) is minimized at φ = 1 and φ = 0. In this case, φ denotes the magnetization; its dynamics are given by this nonconservative partial differential equation, which in the presence of stochastic terms is commonly known as Model A: ∂φ δF 16,17 . On ∂t = −Γ δ φ , where Γ is the mobility of the field the other hand, if φ was a conserved quantity such as concentration, it should have followed a conservative dynamics or Model B given by: ∂∂tφ = Γ∇2 δδF φ . Kinetics terms that cannot be captured by the functional should be added to both types of equations. One such example, in which a kinetic term appears as a nonlinearity, is the Kardar-Parisi-Zhang (KPZ) equation 18 . 1.2
phases the double-well is modified as such: f (φ1 , φ2 , ψ) = aφ12 (1 − φ1 )2 + aφ22 (1 − φ2 )2 + aψ 2 (1 − ψ)2 . Also, the terms accounting for the interaction between phases of the same ma2 1) for the neuterial are added. These are given by: ε1 (∇φ 2
ε1 (∇φ1 )2 ε2 (∇ψ)2 ε3 (∇φ2 )2 + + + 2 2 2 f (φ1 , φ2 , ψ) + H1 φ1 + H2 φ2 ]dV.
Z
[
(2)
The neutrophil and the pathogen both conserve their volumes. Therefore, their dynamics are conservative. Explicitly, their ∂ φ1 2 δF 1 equations are given by dφ dt = γ1 ∇ δ φ = ∂t + V1 · ∇φ1 , and 1
∂ φ2 = γ3 ∇2 δδ F φ2 = ∂t +V2 · ∇φ2 , where γ1 and γ3 are the mobilities, V1 and V2 are the velocities of the neutrophil and the bacteria respectively. We define V1 = b1 ∇c2 and V2 = −b2 ∇c1 in order to guarantee the attraction of the neutrophil by the pathogen and the repulsion of the latter in the process. Here the terms b1 and b2 act as amplifiers to the sensitivity to chemical cues. These now allow us to write the following equations: dφ2 dt
(
∂ φ1 ∂t ∂ φ2 ∂t
= −γ1 ∇2 (ε1 ∇2 φ1 − fφ1 − H1 ) − b1 ∇c2 · ∇φ1 , = −γ3 ∇2 (ε3 ∇2 φ2 − fφ2 − H2 ) + b2 ∇c1 · ∇φ2 ,
(3)
where fφ1 and fφ2 are the derivatives of f with respect to φ1 and φ2 respectively. Now by choosing γ1 ∇2 H1 = −b1 c2 ∇φ1 and γ1 ∇2 H2 = b2 c1 ∇φ1 we ensure that both equations of φ1 and φ2 can be written as continuity equations. Consequently, we impose the conservation of their total volume. Now the dynamics of the leading edge is not conservative as it disappears once the neutrophil reaches the highest chemical ∂ψ δF concentration site 5 . Therefore, dψ dt = −γ2 δ ψ = ∂t +V · ∇ψ, where γ2 is the mobility of the leading edge and V its velocity. This region of the neutrophil moves in synchrony with the rest of the cell and is driven by the chemical attractant c2 . Therefore, we define V = b1 φ1 ∇c2 . The term φ1 appearing in the definition of V couples the motion of the plasma membrane to the rest of the cell. Fig. 1 illustrates the system. The neutrophil secretes c1 . Therefore, it is advected by the same velocity field V1 as φ1 . This in addition to the diffusion and the degradation effects. Similar argument holds for the equation of c2 , which is convected by the same velocity field V2 as the pathogen. The equations governing the dynamics of the system can now be derived by applying the variational derivative to F in Eq. 2 and the results of Eq. 3 to give:
Soft Matter Accepted Manuscript
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DOI: 10.1039/C4SM01842G
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Fig. 1 This figure shows the geometry of the system, reveals the spatial dependence of the fields, and gives a hint on how the velocities are defined. The neutrophil is depicted by φ = 1, the pathogen by φ2 = 1, and the leading edge is shown in red by ψ = 1. The choices of the dependence of the velocities on chemical gradient with proper sign is illustrated.
∂ φ1 2 2 ∂t = −γ1 ∇ (ε1 ∇ φ1 − fφ1 ) − b1 ∇(φ1 ∇c2 ) ∂ ψ ∂t = γ2 (ε2 ∇2 ψ − fψ ) − (b1 φ1 ∇c2 ) · ∇ψ ∂ c1 2 ∂t = D1 ∇ c1 − (b1 ∇c2 ) · ∇c1 − e1 c1 ∂ φ2 2 2 ∂t = −γ3 ∇ (ε3 ∇ φ2 − fφ2 ) + b2 ∇(φ2 ∇c1 ) ∂ c2 = D ∇2 c + (b ∇c ) · ∇c − e c 2 2 2 1 2 2 2 ∂t
(4)
where Di are the diffusion coefficient, ei are the degradation rates (for i = 1, 2 and 3), and fψ is the derivative of f with respect to ψ.
3
Fig. 2 The figure shows snapshots of the dynamics of all three fields φ1 , ψ and φ2 representing respectively the neutrophil, its polymerized region, and the chased cell. They are initially at the lower leftmost part of the domain, and are tracked as they move to its upper rightmost region. Along the way, the neutrophil’s morphological changes can be observed, portrayed by yellow to dark orange cell. Also the evolution of its plasma membrane at its leading edge is depicted by a red color all throughout. The chased cell’s field color ranges from light blue to light green.
Results
We solve the system of Eq. 4 in two dimensions as shown in Fig. 2. The values of γ1 , γ2 , as well as ε1 and ε2 are taken from 13 ; these are experimentally measured parameters. As for the other parameters, unique to our model, we explored a range of values and admissible ones satisfy a condition, which we will derive in the course of our discussion. The neutrophil, its leading edge, and the pathogens are shown at different times during the chase. We notice the amount of deformation the cell undergoes, in addition to the appearance of the leading edge at the left most part of the figure and its disappearance once the pathogen is reached at the rightmost part of the figure, which is consistent with observations. Now we calculate the mean displacement of the center of mass of the neutrophil as a function of time and compare our results with experimental cell motility assays 21,23
and the model given by Parkhurst and Saltzman 22 . It is given by the following equation: D2 (t) = 4µ(t − P + Pe−t/P ), where P is its persistence time and µ is the random motility coefficient. We model the directional motion of cells while 22 models random walks, however both follow the motion of cells in their diffusive states and hence their equivalence. The non-dimensionalised time will allow us to rewrite ′ D2 (t ′ ) = 4µP(t ′ − 1 + e−t ), where t ′ = t/P. Then the product µ × P is essential in determining the significance of our result. In Fig. 3(a) we show the square displacement defined as D2 (t) = (xcm (t) − xcm (t − 1))2 + (ycm (t) − ycm (t − 1))2 , where xcm and ycm are the coordinates of the neutrophil’s center of mass. D2 (t) will decrease as the cell approaches the pathogen. Therefore, we truncate it to exclude the regime where the neutrophil slows down, as shown in Fig. 3(b). The conversion from pixels to µm is done by choosing the diameter of the cell in pixels to correspond to d = 8.5±0.5µm, which is the average diameter of a neutrophil. The numerical scheme used in solving Eq. 4 has an error of the order of h2x , where the latter is the grid step size. The square displacement computed from the numerical solution of our model then has an error of the order of h4x . The equation of D2 , fitted to that extracted from the numerical simulation, is shown in red in Fig. 3 and predicts a persistence time of 1–6 | 3
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DOI: 10.1039/C4SM01842G
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DOI: 10.1039/C4SM01842G
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Fig. 4 The figure shows the evolution of system’s components. The position of the neutrophil with respect to that of the pathogen and the level curve c2 of the repellant is followed.
(b)
Fig. 3 In (a) the square of the displacement of the neutrophil’s center of mass as a function of iteration number is shown, whereas in (b) the neutrophil is tracked before reaching the pathogen and the truncated displacement of (a) is plotted. The red curve is the fit of the model given by: D2 (t) = 4µ(t − P + Pe−t/P ). Also, the numerical scheme used to solving Eq. 4 and thus determining the position of the neutrophil has an error of the order of h2x , where hx is the grid step size. This defines the error bar on the square displacement to be h4x .
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Finally, we can make a prediction for the condition under which the neutrophil will fail to neutralize the pathogen and the immune response is then inefficient. We take the limit where the degradation terms are slow compared to the convective ones and thus set e1 and 2 to 0. Then c1 and c2 are now Gaussian advected by the flow and are centered at the cell’s and pathogen’s centers of mass respectively. Fig. 4 is an illustration of the dynamics. The new position of the center of mass of the pathogen after one iteration is given by dc1 1 (x p , y p ) = (x p0 − b2t dc dx , y p0 − b2 t dy ), where x p0 and y p0 are the initial pathogen’s center of mass coordinate, and t is the time step size. Similarly, the neutrophil in this time step has dc2 2 moved (xn , yn ) = (xn0 + b1t dc dx , yn0 + b1 t dy ), where xn0 and yn0 are the initial coordinates of the neutrophil’s closest point
Soft Matter Accepted Manuscript
P = 1 [iteration] and a motility µ = 19.2 × 10−8 [µm2 per iteration]. For a collagen concentration of 0.1mg/ml, for example, the motility is µexperimental = 0.51 ± 0.04µm2 /s and Pexperimental = 40 ± 25s 22 . Their product A = 20 ± 15[µm2 ]. This interval is in agreement with our numerical results, where µ × P = 19 ± 1[µm2 ]. For different collagen concentrations the value of P is still consistent with the experimentally measured values. This shows that the model is capable of reproducing physically measurable properties of the system and confirms the validity of the assumptions made about the velocity dependence on the chemical gradient.
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√ to the pathogen. c2 , which is a Gaussian of width w = 4D2t, defines a level curve in which the neutrophil must partially always lie. This is translated by the following inequality: (5)
which should hold for all times. This condition can be put to experimental test to check its validity and confirm our findings. In 1-D, Eq. 5 then becomes the difference between the neudc2 1 tropil and pathogen’s velocities: b2 dc dx + b1 dx = vn − v p > √ x p0 −xn0 −2 D2 t . t
In Fig. 5 we show, for the 1D motion, the region of parameter space where the immune response is successful. In 2D the condition is defined in a 4 dimensional parameter space.
2
1.5
1
Fig. 5 The plot shows the condition for a successful immune response relating the difference in the neutrophil’s and pathogen’s velocities versus the difference in their initial positions. In the area above the curve the immune response is effective.
Neutrophil s Decision
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(x p − xn )2 + (y p − yn )2 < w2 ,
Further, the neutrophil is affected by intermediary chemoattractants such as IL−8 and LTB4 , which are encountered during its migration to the site of infection 27 . Therefore, the need to prioritize in the presence of competing signals becomes vital. We establish a dose-response curve in the case of two conflicting chemoattractants by varying the distance between them as well as their corresponding cues’ sensitivities, or velocity amplifications b11 and b12 . We follow the decision of the neutrophil and assign the value 1 in case it migrates to site 1 and −1 otherwise. Fig. 6 suggests that the neutrophil is distracted by intermediary cues; that is when it chooses to migrate to site 2. This is in total agreement with experimental observations of Pten−/− neutrophils 28 . However, the neutrophil can selectively ignore these cues using PTEN-dependent mechanism regardless of their concentrations 28 . Therefore, at each time step we rescale our parameter b11 in an adaptive way to achieve directional migration to site 1 inhibiting the effect of other cues and replicating the effect of the PTEN mechanism while otherwise fixing b11 would recover the distracting effect of intermediary cues in Pten−/− neutrophils.
0.5
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Therefore, modifying b1 , which is the velocity amplification factor of the neutrophil, could have drastic effects on the dynamics. Explicitly, the effect of chemotherapy on breast cancer is known to lead to a failure in neutrophil chemotactic function, specifically to stimuli like fMLP and LTB4 26 . In other words, a pathogen would escape neutralization in sick individuals undergoing chemotherapy while in healthy individuals that same pathogen would be successfully neutralized. This reduced migratory capacity of the neutrophils after chemotherapy increases the risk of infection, which our model predicts when b1 is reduced in accordance with reported experimental findings 26 , which establishes a relation between the parameter b1 and the absence of immunosuppressive factors.
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Soft Matter Accepted Manuscript
DOI: 10.1039/C4SM01842G
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Fig. 6 The plot shows the decision of the neutrophil to go to either site 1 or site 2 as a function of the chemical cues’ sensitivities and the distance between the sites. 1 corresponds to migrating to site 1 and −1 to site 2.
1–6 | 5
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DOI: 10.1039/C4SM01842G
Conclusions
In this paper, we have used the phase-field method to replicate an immune response. We have reproduced the membrane deformations the neutrophil undergoes as it responds to chemical cues. In addition, we have derived an inequality that relates the speed of the neutrophil to that of the pathogen and the diffusion coefficient of the chemical attractant that the latter secretes. This is a consequential result that models how the immune system succeeds in fighting the pathogens or fails based on three major parameters, the pathogens and the neutrophil’s sensitivities to chemical cues, and the diffusion coefficient of the attractant. Tuning these parameters is key to a successful immune reaction. Additionally, in contrast to experimental setups, which under agarose neutrophil migration assays are either unidirectional or establish a limited number of chemical fields, we can generate complex chemotactic fields and develop numerical single cell assays that would help in pharmaceutical treatments. We have presented an additional application to phase-field in the context of active matter 25 and proved its efficiency in reproducing salient features of the dynamical processes at play.
5
Acknowledgments
We acknowledge support from the Natural Sciences and Engineering Research Council of Canada and le Fonds de recherche du Qu´ebec — Nature et technologies.
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P.A. Iglesias and P.N. Devreotes, Curr. Opin. Cell Biol., 2008, 20, 35. C.A. Parent and P.N. Devreotes, Science, 1999 284, 765. G. William et al., J. Leukocyte Biol., 2006, 79. L.V. Dekker and A.W Segal, Science, 2000, 287, 982. M. Eisenbach. Chemotaxis (Imperial College Press, 2004). A. Gamba et al., Proc. Natl. Acad. Sci., 2005, 102, 16927 P. Friedl and D. Gilmour. Nature Reviews Molecular cell biology, 2009, 10, 445. L. Benov and I. Fridovich. Proc. Natl. Acad. Sci., 1996, 93, 4999. B. Alberts et al., Molecular Biology of the Cell (Garland Science 2002). N.S Gov. HFSP Journal,2009, 3. Yangi et al., BMC Syst. Biol.,2008, 2, 68. C.W. Wolgemuth and M. Zajac, J. Comput. Phys., 2010, 229, 7287. D. Shao, H. Levine, and W.J. Rappel, Proc. Natl. Acad. Sci., 2012, 109, 6851. S. Najem and M. Grant. Phys. Rev. E , 2013, 88, 034702. D. Ruelle, Statistical Mechanics: Rigorous Results (New York: W.A. Benjamin Inc,1969) N. Provatas and K. Elder, Phase Field Methods in Materials Science and Engineering (WILEY-VCH, 2010). Chaikin P.M, Lubensky T.C. Principles of Condensed Matter (Cambridge University Press, 1995) M. Kardar, G. Parisi, and Y.C Zhang, Phys. Rev. Lett., 1986, 56, 889. J.E Gestwicki and L.L Kiessling, Nature, 2002, 415, 81.
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20 E.F Keller and L.A Segel, J. Theoret. Biol, 1971, 30, 225. 21 P.V Moghea, R.D Nelsonb, and R.T Tranquillo, J Immunol. Methods, 1995, 180, 193. 22 M.R Parkhurst and W.M Saltzman, Biophys. J., 1992, 61, 306. 23 F. Lin et al., Biochem. Bioph. Res. Co., 2004, 319, 576. 24 S.H Zigmond, Nature, 1974, 249, 450. 25 S. Najem and M. Grant, Europhys. Lett., 2013, 102, 16001. 26 M.A, Mendona et al, Cancer chemotherapy and pharmacology, 2006, 57 (5), 663. 27 B. Heit, Bryan et al.,The Journal of cell biology, 2002, 159, 91. 28 B Heit et al., Nature immunology, 2008, 9, 743.
Soft Matter Accepted Manuscript
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We model the interplay between a neutrophil and a pathogen, follow their morphodynamics, and identify a condition for the success of the immune response.
Soft Matter Accepted Manuscript
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TABLE OF CONTENTS
1
