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We present a reaction–diffusion mathematical model for the evolutionary dynamics of phenotypic evolution. A detailed deduction of the equations is presented for the one-dimensional version, from which a more general model is proposed. Particular cases are studied using analytical approximations and numerical simulations. Results indicate that the approach proposed produces results that are coherent with mainstream models in evolutionary dynamics, suggesting that the reaction–diffusion model could be an alternative tool in the analysis of evolutionary dynamics.

Although essential for inflammatory responses, leukocyte recruitment to blood vessel walls in response to inflammatory stimuli, such as TNF‐α, can contribute to vascular occlusion in inflammatory diseases, including atherosclerosis. We aimed to further characterize the mechanisms by which TNF stimulates adhesive and morphologic alterations in neutrophils. Microfluidic and intravital assays confirmed the potent effect that TNF has on human and murine neutrophil adhesion and recruitment in vitro and in vivo, respectively. Inhibition of actin polymerization by cytochalasin D significantly diminished TNF‐induced human neutrophil adhesion in vitro and abolished TNF‐induced membrane alterations and cell spreading. In contrast, TNF‐induced increases in β2‐integrin (Mac‐1 and LFA‐1) expression was not significantly altered by actin polymerization inhibition. Consistent with a role for cytoskeletal rearrangements in TNF‐induced adhesion, TNF augmented the activity of the Rho GTPase, RhoA, in human neutrophils. However, inhibition of the major RhoA effector protein, Rho kinase (ROCK), by Y‐27632 failed to inhibit TNF‐induced neutrophil adhesion. In contrast, the formin FH2 domain inhibitor, SMIFH2, abolished TNF‐induced human neutrophil adhesion and diminished leukocyte recruitment in vivo. SMIFH2 also inhibited TNF‐induced cytoskeletal reorganization in human neutrophils and abolished the alterations in β2‐integrin expression elicited by TNF stimulation. As such, Rho GTPase/mDia formin‐mediated cytoskeletal reorganization appears to participate in the orchestration of TNF‐induced neutrophil‐adhesive interactions, possibly mediated by formin‐mediated actin nucleation and subsequent modulation of β2‐integrin activity on the neutrophil surface. This pathway may represent a pharmacologic target for reducing leukocyte recruitment in inflammatory diseases.

Sickle cell anemia (SCA) is a chronic inflammatory disease, in which activated neutrophils play a role in initiating vaso-occlusion. Two populations of circulating neutrophils have been described, denominated as low-(LDN) and high-(HDN) density neutrophils. Circulating numbers of LDN (a less inflammatory subset) are normally minimal, but this population augments under inflammatory stress, such as that seen in cancer. Transforming growth factor beta-1 (TGF-β1) is a cytokine with anti-inflammatory properties that is elevated in SCA. In conditions such as Chron's disease, TGF-β1 has protective effects, mediated by its immuno-suppressive functions. In macrophages, it is thought to trigger the polarization from pro- (M1) to anti-inflammatory phenotypes (M2), which hypothetically occur in neutrophils too (N1 and N2). Moreover, dimethyl sulfoxide (DMSO) reportedly increases TGF-β receptors expression on epithelial cells. We aimed to characterize the subsets of circulating neutrophils in SCA patients and investigate the effects of TGF-β1 and DMSO on these cells. Neutrophils from healthy (CON) and SCA individuals, in steady state and without blood transfusion for 90 days, were isolated from peripheral blood by Ficoll-Paque density gradient centrifugation. HDN and LDN were obtained from the granulocyte and mononuclear layers, respectively, and stained with CD66b for neutrophil identification by flow cytometry. As no significant effect of hydroxyurea (HU) therapy on the data obtained was observed, patients' data were grouped together irrespective of HU use. Percentages of LDN were calculated based on the total of gated CD66b+cells. SCA patients had higher levels of LDN than CON (3.2±0.9%, N=7 vs1.3±0.3%, N=13; p=0.02). We next investigated the presence of CD66b+/CD206-and CD66b+/CD206+cells, to infer the presence of N1 and N2 phenotypes, respectively. N2 were more frequent in the LDN than in the HDN subset (CON: 68.1±3.3% vs52.0±4.4%, N=9, p=0.01; SCA: 77.6±8.9% vs44.1±5.0%, N=3, p=0.03). To determine whether TGF-β1 and DMSO could shift HDN to a LDN profile, cells were treated (2h) with TGF-β1 (50pM) and/or DMSO (1.5%). Treatments with DMSO alone or combined with TGF-β1 increased the percentage of CD206+cells in CON (45.7±2.1% vs61.9±7.6 and 53.6±2.6% respectively, N=6, p=0.04), as well as CD206+expression on each cell (mean fluorescence intensity, MFI) (137.5±16.9 MFI vs293.6±71.2 MFI and 210.1±23.9 MFI, respectively, p=0.03). In SCA, only the combined TGF-β1/DMSO treatment increased the MFI of CD206 in HDN (115.7±10.2 vs255.8±29.7 MFI, N=4, p=0.03). We next investigated whether TGF-β1/DMSO could reduce the adhesion of HDN to fibronectin ligand (FN, 20μg/mL) using static adhesion assays (30 min, 37ºC). HDN from CON and SCA were treated with TGF-β1 and/or DMSO (90min) and stimulated with TNF-α (200ng/mL, 30min). Although TGF-β1 alone did not reduce the adhesion of HDN to FN (p>0.05), the addition of DMSO decreased TNF-α-induced adhesion in CON (16.5±1.8% to 11.3±1.5%, p=0.03, N=10) and SCA HDN (38.9±23.9% to 13.9±1.5%, p=0.04, N=3). Subsequently, HDN were stimulated (4h) with LPS (100ng/mL) and INF-γ (20ng/mL), to induce N2 polarization, in the presence/absence of TGF-β1 and DMSO. The combined treatment again reduced adhesion in both groups (CON: 11.1±1.5% to 4.2±1.2%, p<0.01, N=4; SCA: 47.4±12.3% vs 21.9±4.5%, p=0.04, N=4). To assess whether TGF-β1 and DMSO could affect the production of proinflammatory cytokines by HDN after stimulation with LPS/INF-γ, TNF-α and IL-1β levels in cell supernatants were measured by ELISA. TGF-β1 and DMSO, in combination, decreased both TNF-α and IL-1β release from CON (TNF-α: 39.7±8.7pg/mL to 8.7±0.9pg/mL, p<0.01, N=6; IL-1β: 66.9±10.5pg/mL to 16.5±5.1pg/mL, p=0.02, N=4) and SCA HDN (TNF-α: 174.0±55.5pg/mL to 21.8±6.6pg/mL, p=0.01, N=8; IL-1β: 103.6±25.6pg/mL to 43.1±15.5pg/mL, p=0.01, N=8). Our results demonstrate for the first time the presence of elevated numbers of LDN in SCA patients, indicating an increased basal response to inflammatory stress. However, this shift in the anti-inflammatory subset does not appear enough to control inflammatory responses in the disease, and the use of agents capable of inducing this polarization may be a promising approach. Moreover, the anti-inflammatory effects of TGF-β1 on HDN seem to be enhanced by DMSO and suggest this combination as an effective modulator of the inflammatory profile of neutrophils.

No relevant conflicts of interest to declare.

Background:Intravascular hemolysis, a major complication of sickle cell anemia and malaria among other diseases, incurs the release of excessive quantities of hemoglobin and heme from red blood cells. If not adequately sequestered by hemoglobin- and heme-binding proteins, these molecules may incur oxidative stress, thrombosis and cellular activation. Heme is now recognized as a red cell DAMP (damage-associated molecular pattern) that stimulates inflammasome formation in macrophages and can induce neutrophil extracellular trap release under certain circumstances. Aim:This study evaluated the in vitroeffects of heme on the adhesive properties of human neutrophils. Methods:Neutrophils were separated from the peripheral blood of healthy individuals and their adhesion in the presence/absence of heme was compared by static adhesion assays using myeloperoxidase, for quantification of cell adhesion (30 min, 37oC, 5% CO2). Results:Heme (50 µM) significantly increased the adhesion of neutrophils to fibronectin (FN) and to recombinant ICAM-1 (an endothelial ligand), when compared to non-treated neutrophils (43.7±4.6%; 10.9±21.4 %, respectively, n=5, P<0.001 for FN and 38.1±3.6%; 6.9±0.8%, respectively, n=6, P<0.001 for ICAM-1). Interestingly, heme induced neutrophil adhesion even more efficiently than the potent pro-inflammatory cytokine, TNF-α (200 ng/ml) (35.5±5.2% adhesion to FN, n=6, P<0.05). Furthermore, inhibition of the NFκB transcription factor with the pharmacological inhibitor, BAY 11-7082 (20 μM), abolished heme-stimulated neutrophils adhesion to FN (reduced from 41.8±6.7% to 7.4±0.5%. n=6; P<0.001). Flow cytometry demonstrated that while TNF-α significantly increases the expression of the Mac-1 integrin subunit, CD11b (data not shown, P<0.05), but not the LFA-1 integrin subunit CD11a (data not shown, P>0.05), on the surface of neutrophils, heme did not augment CD11b or CD11a expression (P>0.05). In contrast, heme significantly augmented the active conformations of these two β2 integrin subunits, as demonstrated by epitope-specific antibodies (CD11b; heme: 520.5±51.8; basal 123.7±16.7 MFI, n=10, P<0.001 and CD11a; heme: 69.8±3.9; basal: 43.8±2.0 MFI, n=6, P>0.001). Inhibition of NFκB translocation with BAY 11-7082 (20 μM) significantly decreased the activity of the LFA-1 integrin on the surface of neutrophils after heme stimulation (reduced to 47.6±4.2 MIF, n=6; P<0.05). To assess whether heme-induced neutrophil adhesive properties are mediated by cytoskeletal rearrangements, we evaluated the effects of cytochalasin D (0.5 μg/ml), an inhibitor of actin polymerization. While cytochalasin D inhibited TNF-α-induced neutrophils adhesion (data not shown, P<0.05), this compound did not significantly alter the adhesive properties of heme-stimulated neutrophils to FN (heme: 22.8±2.3%, p<0.001; cytochalasin D: 22.5±2.0%, p=0.4; n=6). Pre-incubation of heme-stimulated neutrophils with the antioxidants, ascorbic acid (120 µM; 3.5 hours) and α-tocopherol (1mM; 15 min), reduced their adhesion to FN by 14.4±9.2% and 46.5±5.5% respect. n=6, p<0.05). Conclusion:We therefore demonstrate, herein, that heme is a potent activator of neutrophil adhesive properties, increasing the ligand affinity of the β2 integrins via a mechanism that is apparently mediated by an NFkB-dependent pathway. The mechanism of neutrophil activation appears to differ from that stimulated by TNF-α and may involve, in part, the generation of reactive oxygen species. Given the fundamental role that the adhesion of neutrophils to the vascular wall plays in the vaso-occlusive process in sickle cell disease and other vascular inflammatory processes, our findings further support the idea that cell-free heme represents a major therapeutic target in the hemolytic diseases.

Mathematical models describing the behavior of hypothetical species in spatially heterogeneous environments are discussed and analyzed using the fibering method devised and developed by S. I. Pohozaev.