Abstract
Psoriasis is a chronic inflammatory skin disease clinically characterized by the appearance of red colored, welldemarcated plaques with thickened skin and with silvery scales. Recent studies have established the involvement of a complex signalling network of interactions between cytokines, immune cells and skin cells called keratinocytes. Keratinocytes form the cells of the outermost layer of the skin (epidermis). Visible plaques in psoriasis are developed due to the fast proliferation and unusual differentiation of keratinocyte cells. Despite that, the exact mechanism of the appearance of these plaques in the cytokineimmune cell network is not clear. A mathematical model embodying interactions between key immune cells believed to be involved in psoriasis, keratinocytes and relevant cytokines has been developed. The complex network formed of these interactions poses several challenges. Here, we choose to study subnetworks of this complex network and initially focus on interactions involving \(\hbox {TNF}_{\alpha }\), IL23/IL17, and IL15. These are chosen based on known evidence of their therapeutic efficacy. In addition, we explore the role of IL15 in the pathogenesis of psoriasis and its potential as a future drug target for a novel treatment option. We perform steady state analyses for these subnetworks and demonstrate that the interactions between cells, driven by cytokines could cause the emergence of a psoriasis state (hyperproliferation of keratinocytes) when levels of \(\hbox {TNF}_{\alpha }\), IL23/IL17 or IL15 are increased. The model results explain and support the clinical potentiality of anticytokine treatments. Interestingly, our results suggest different dynamic scenarios underpin the pathogenesis of psoriasis, depending upon the dominant cytokines of subnetworks. We observed that the increase in the level of IL23/IL17 and IL15 could lead to psoriasis via a bistable route, whereas an increase in the level of \(\hbox {TNF}_{\alpha }\) would lead to a monotonic and gradual disease progression. Further, we demonstrate how this insight, bistability, could be exploited to improve the current therapies and develop novel treatment strategies for psoriasis.
Introduction
Psoriasis is a prevalent chronic inflammatory skin disease which affects people of all ages and ethnicities^{1}. Chronic plaque psoriasis (also known as psoriasis vulgaris) is the commonest form and occurs in 90% of cases^{2}. It has a characteristic appearance that consists of well demarcated, inflamed plaques covered with silverywhite scales. The outermost layer of the human skin (epidermis) thickens in these lesions due to increased proliferation and abnormal differentiation of keratinocytes, which are the predominant cells of the epidermis. The severity of psoriasis can range from the involvement of small, isolated areas to covering the majority of patients’ skin. In the case of mild disease, topical treatments are prescribed whereas phototherapy (using ultraviolet light), systemic and biological therapies are used for moderatetosevere cases. Although these therapies are often successful, response varies from patient to patient, and it is therefore crucial to better understand why these treatments work in some cases and not others.
Psoriasis had previously been described as a primary disorder of keratinocyte proliferation and differentiation. However, advances in genetic and immunological techniques have led to improvements in our understanding of psoriasis, which is now widely recognised as a complex and multifactorial, immunemediated inflammatory disease^{3,4,5}. Psoriasis can therefore be thought of as a multiscale dynamic disease arising as a manifestation of the dysregulated interaction between keratinocytes and the immune system. Given this complexity and in order to understand how the disease state arises, research strategies involving experimental and mathematical modelling of these processes are essential^{6}. As such, several in vitro and in vivo models have been proposed to study its underlying causes^{7} along with the use of in silico models to understand healthy and psoriatic epidermic growth. These in silico models can be classified into two groups: (1) agentbased^{8,9,10,11,12}, and (2) ordinary differential equation (ODE) based^{13,14,15}. The agent based models could simulate the nonlesional and psoriatic keratinocyte compartment of the epidermis as well as the UVB (Ultraviolet B) intervention for plaque resolution. In this method, each cell is represented by a software agent and each agent has a rule set that determines the behavior. The rule set also determines the interaction of agents with their neighbours and that can be used to model the organisation of multicellular aggregates^{8}. An ODE model developed by Valeyev et al.^{13} considered interactions between immune cell populations based on their cytokine production profiles and informed the mechanism of inflammation in human skin. Mathematical models for the pathogenesis of psoriasis have been developed by Roy et al.^{14,15} involving the densities of immune cells and keratinocytes. They have considered the hyperproliferation of keratinocytes as one of the precursors of psoriasis. Further, they demonstrated a method for stable control of the keratinocyte population using negative feedback. In a recent study, they have demonstrated the role of Th1 and Th2 cells in the pathogenesis of psoriasis using an ODE based mathematical model^{16}.
In order to further our understanding of the role that immune cells and cytokines play in the pathogenesis of psoriasis, we recently synthesised current evidence into a mathematical model of the network of cytokinemediated interactions between key immune cells (Tcells and dendritic cells) and keratinocytes^{18}. The inclusion of keratinocytes was crucial to enable the diseased and healthy psoriasis phenotypes to be modelled. Within this framework, we demonstrated the existence of healthy and diseased states, as well as the potential for cytokines to behave as finitetime controllers of the state of the system. This model therefore offers the potential to advance our understanding of the mechanisms underlying the success or failure of biological treatments. A challenging task in terms of analysis, however, is the complexity of the system, as interactions between cells are mediated by many different cytokines.
To further our insight into this system here we identify specific cytokines those mediate subnetworks of the network formed of key immune cells and keratinocytes. Such subnetworks could be particularly important for governing system dynamics^{19}, but also allow a systematic analysis of how they may mediate transitions between healthy and psoriatic states. In particular, we focus on subnetworks mediated by cytokines \(\hbox {TNF}_{\alpha }\), IL17/IL23 axis and IL15. Selection of \(\hbox {TNF}_{\alpha }\) and IL17/IL23 axis is inspired by biologics either approved by the FDA (Food and Drug Administration) or under clinical trials targeting them^{20,21,22}. However, to date there is no psoriasis treatment directly targeting IL15. Indeed, evidence linking IL15 to psoriasis pathogenesis is less abundant than the other cytokines considered above (though see, for example^{23,24,25}). Nevertheless, the interactions between cell types relevant for psoriasis that we identified do involve IL15 and our research indicates that IL15 links all 3 cell types considered. It is therefore interesting to understand the role that it plays in regulating the dynamics of this network. In addition, the structure of the subnetwork mediated by IL15 resembles the well known coherent feed forward loop motif observed and analysed in the context of gene regulatory networks .We study the asymptotic behaviour of these subnetworks and demonstrate the potential for each of these cytokines to mediate transitions between healthy and psoriasis states. We find that increases in levels of \(\hbox {TNF}_{\alpha }\) and IL23/IL17 lead to a high keratinocyte state, which is in line with their inhibitors being used as treatments for psoriasis. Our analysis further suggests that these cytokines exhibit different dynamic routes to psoriasis. Hyperproliferation of keratinocytes due to increases in the level of IL23/IL17 can occur via a bistable regime in which two steady state populations of keratinocytes is possible. A similar behaviour also occurs for the subnetwork involving cytokines IL15. In contrast, the progression of psoriasis in the subnetwork formed by \(\hbox {TNF}_{\alpha }\) is found to occur without bistability, instead giving rise to a monotonic increase in the keratinocyte population. Due to bistability, we expect the possibility of a quicker disease progression mediated by IL17/IL23 and IL15 than \(\hbox {TNF}_\alpha\). Thus, we may observe a faster onset of recovery for biologics targeting IL17/IL23 and IL15 in compared to \(\hbox {TNF}_\alpha\) inhibitors. These results are consistent with the observation that IL17 inhibitors act quickly and efficiently than \(\hbox {TNF}_\alpha\) inhibitors^{21,22,26}. However, the better clinical response of IL17 is not attributed to bistability yet. Further, we discuss the relevance of these findings for the future development of treatments for psoriasis.
Results
Dynamics of the subnetwork mediated by the \(\hbox {TNF}_{\alpha }\)
We first study the dynamics of the subnetwork that emerges due to interactions mediated by \(\hbox {TNF}_{\alpha }\) (Fig. 1B; Eq. 7). Although we have greatly reduced the complexity of the system, six parameters remain which determine its dynamics.
The steady state behaviour of the subnetwork for varying model parameters is investigated using bifurcation analysis techniques of dynamical systems and the result is shown in Fig. 2. We study the effect of varying parameters independently in the first instance, focussing on parameters that directly modulate the keratinocyte population (such as q), embody the cytokine effect (\(b^{'}_{TNF}\)) and the effect of changes in Tcell numbers (\({\bar{L}}\)). We observe that an increase in the modulatory effect of \(\hbox {TNF}_{\alpha }\) i.e. high level of \(TNF_{\alpha }\) represented by \(b^{'}_{TNF}\) could lead to increase in the population of keratinocytes and therefore to psoriasis as the keratinocytes are observed to be hyperproliferative in psoriatic skin^{3,27,28}. The keratinocyte population would also increase with increase in the effective rate of migration of keratinocytes denoted by q. Also, psoriasis would develop when the strength of the modulatory effect of dendritic cells on other modelled cells is stronger than that of keratinocytes, represented by a large value of u (Fig. 2A). The large vale of u (= \(\frac{k_D}{k_K}\)) suggests that the impact of the keratinocyte cell population on dendritic cells and its own population would be stronger than the impact of the dendritic cell population.
In contrast to the above, the keratinocyte population decreases with increase in the steady state population of Tcells (\({\bar{L}}\)), Hill coefficient (n), and effective rate of migration of dendritic cells towards the lesion (p) (Fig. 2).
Based on the known mode of action of antiTNF biologics, we use Fig. 2A to define healthy and psoriasis states. It is henceforth assumed that a value of \({\bar{K}}\) above 4 is considered as a psoriasis state. In fact, a fourfold increase in the number of keratinocytes compared to healthy skin has been previously suggested^{13}. We observe a gradual and monotonic increase in keratinocytes therefore these results suggests that this subnetwork would lead to a gradual progression of psoriasis.
We extend this analysis by considering the interplay between \(b^{'}_{TNF}\) and the other parameters in Fig. 2B–F. Figure 2B demonstrates that decreasing the value of the Hill coefficient, n in Eq. (7), causes keratinocyte levels to have a higher sensitivity to increases in \(b^{'}_{TNF}\). This sensitivity would also increase with increase in the effective rate of inward migration of keratinocytes (Fig. 2B) and the modulatory effect of the dendritic cell population being more effective than that of the keratinocyte population (Fig. 2C). Nonetheless, this sensitivity would be decreased by increasing the steady state level of Tcells (\({\bar{L}}\)) and the inward effective rate of migration of the dendritic cell population (p) (Fig. 2E, F). Here, \(k_D\) and \(k_K\) are assumed to have the properties of a dissociation constant in a Hill function. These results suggest that the steepness in increase or decrease in the population of keratinocytes in response to varying level of \(\hbox {TNF}_{\alpha }\) could be modified by changing other model parameters. In other words, the sensitivity of the increase or decrease in the keratinocyte population to fluctuating level of \(\hbox {TNF}_{\alpha }\) would be determined by other model parameters.
Dynamics of the IL23 and IL17 mediated subnetwork
This subnetwork is mediated by the widely known “IL23/IL17” axis which has been extensively explored for developing a treatment for psoriasis^{22,29,30}. This subnetwork is formed of two feedforward loops between dendritic and Tcells, the output of which determines the dynamics of the population of keratinocytes (Fig. 1C). Alongwith studying the asymptotic behavior of this subnetwork, we wish to determine whether the net effect of the output of this axis on the population of keratinocytes is enhancing or inhibitory. We performed a steady state analysis of the subnetwork equations, accounting for either an activating (case 1) or inhibiting (case 2) effect of dendritic and Tcells on the population of keratinocytes (governed by Eqs. 9 and 11, respectively).
A steady state characteristic behavior of this subnetwork in case 1 is shown in Fig. 3 and Fig. S1A–L. Figure 3A suggests that the keratinocyte population increases with increase in the level of either of the cytokines, IL23 and IL17, represented by parameters \(b^{'}_{IL23}\) and \(b^{'}_{IL17}\), respectively. However, the increase is more sensitive to changes in IL17 compared to IL23 (Fig. 3A) . The keratinocyte population also increases with increase in the parameters p, q and r representing the effective rate of migration of dendritic, keratinocytes and Tcells, respectively towards the lesion. Increases in the parameter \(k_A\) and n (representing the effective dissociation constant and Hill coefficient, respectively for the combined effect of IL23 and IL17) leads to a decrease in the population of keratinocytes. Note that changes in the above mentioned parameters result in either increase or decrease in the population of keratinocytes but monotonically. Therefore, these results suggest that for some parameter combinations, changes in the model parameters could lead to a gradual increase or decrease in the keratinocyte population.
Interestingly, the changes in the modulatory effect of IL23 (\(b^{'}_{IL23}\)) or IL17 (\(b^{'}_{IL17}\)) could lead to a pair of saddlenode bifurcations, and hence a region of bistability (Fig. 3B,C) especially for \(n>1\). The bistable region is bounded by the saddle node bifurcations, \(74.5<b^{'}_{IL23}<104.3\) and \(0.4<b^{'}_{IL17}<25\) for the given parameter combinations and denoted by LP in the figure. In fact, there is a large region of bistability in the two parameter spaces of n vs \(b^{'}_{IL23}\) and n vs \(b^{'}_{IL17}\) as depicted in Fig. 3D and Fig. S1K,L. In the bistable region, two stable and an unstable steady state, shown by the solid and dotted line in Fig. 3B,C, respectively coexist. The bifurcation points separate the system’s response into two distinct regions—low and high keratinocyte populations. The meaning of the bistability will be further explained in the next section and Fig. 5. Note that, there is \(\approx\) 4 fold difference in the low and high keratinocyte levels similar to the previous subnetwork (Fig. 1B). Also the high and low level of keratinocytes correspond to a psoriatic (diseased) and healthy state, respectively.
Our analysis suggests that the increase in the population of keratinocytes in response to the increase in the level of IL17 would be sensitive to the present level of IL23 (Fig. 3E). In fact, if the present level of IL23 is comparatively high then the changes in keratinocyte population would be more sensitive to the changes in the level of IL17 (Fig. 3E). Similarly, the present level of IL17 would determine the sensitivity of the increase in the population of keratinocytes in response to the increase in the level of IL23 (Fig. 3F). Like in panel E of Fig. 3 , a high present level of IL17 ensures comparatively high sensitivity of increases in keratinocytes, in response to changes in the level of IL23.
The steady state behavior of the subnetwork in case 2 is shown in Fig. 4. As shown in Fig. 4A, in contrast to case 1, the population of keratinocytes decreases with an increase in the level of IL17 (\(b^{'}_{IL17}\)) and IL23 (\(b^{'}_{IL23}\)) as well as with increase in the effective rate of migration of the cell populations (represented by p, q, and r). However, the high sensitivity to changes in the level of IL17 compared to that of in the level of IL23, is preserved as the population of keratinocytes decreases faster with increase in IL17. Therefore, our analysis suggests that the proliferation of keratinocytes would be more sensitive to changes in the level of IL17 than to IL23 (Figs. 3A, 4A). Because of this, we argue that an IL17 blocker could potentially be more effective than an IL23 inhibitor.
The population of keratinocytes increases with increase in the parameter \(k_A\) and n, which is opposite to what is observed in case 1. Therefore, the response to changes in the level of IL17 and IL23 in cases 1 and 2 are antagonist to each other (Figs. 3A, 4A) as expected.
In addition, compared to IL17 and IL23, the increase in the keratinocyte population is less sensitive to changes in the net population of dendritic and Tcells for small values of \(b_{net}\) (Fig. S1B, G). For large values of \(b_{net}\), the keratinocyte population increases almost linearly (Fig. S1B,G). This highlights the multifaceted impact of changes in the cytokine levels in comparison with the effects of direct changes in the cell population.
We observe that case 2 could also give rise to a region of bistability in response to the changes in the level of IL17 and IL23 (Fig. 4B,C). However, in this case, for low levels of IL23 and IL17 the population of keratinocytes would be very high which is opposite to the case 1. In this case too, the region of bistability is large (Fig. 4F, Fig. S2L–N).
As with case 1, the decrease in the population of keratinocytes in response to increase in the level of IL17 is sensitive to the present levels of IL23 (Fig. 4E). In fact, it decreases faster at high level of IL23 in response to the increase in the level of IL17. Similarly, the population of keratinocytes decreases faster if the present level of IL17 is high in response to an increase in IL23 (Fig. 4F). These results suggest that although, in case 2, the population of keratinocytes decreases with increase in the level of IL17 and IL23, as opposed to case 1, the sensitivity of decrease (increase in case 1) would still be determined by the present level of IL23 and IL17, respectively.
Our analysis demonstrates that the net effect of the populations of dendritic and Tcells on keratinocytes would be activating to be consistent with the current knowledge about IL17. The IL17 family of cytokines have been suggested to be most relevant to the pathogenesis of psoriasis because of their action on keratinocytes and the increased level within psoriatic lesions^{31,32}. Also, higher serum and lesional levels of a member of this family, IL17A, in psoriasis patients compared to controls have been reported which supports the role of IL17 A in this disease^{31,33}. In addition, the use of IL17 blockade (ixekizumab) has shown a significant reduction in keratinocyte proliferation and infiltration of Tcells and dendritic cells into the dermis and epidermis compared to the baseline^{34}. Therefore, the hypothesis of case 1 is most likely in in vivo conditions.
Dynamics of the subnetwork involving IL15
This subnetwork is formed of a coherent feedforward loop and self activation. Following the mathematical modelling frame work discussed above the population dynamics of this subnetwork (Fig. 1D) is governed by Eq. (14). The steady state response of the keratinocyte population to varying model parameters independently is shown in Fig. 5A. We observe that the increase in the level of IL15, denoted by the large value of \(b^{'}_{IL15}\) would lead to increased keratinocyte population and therefore a psoriatic state. Similarly, increase in the parameter q and r representing the effective rate of migration of keratinocytes and T cells, respectively towards the lesion would result in psoriasis.
The keratinocyte population would also increase with increase in the parameter \(v (=\frac{k_d}{k_k})\) which compares the strength of the effect caused by dendritic cells and keratinocytes on other modelled cell types (Fig. 5A). A large value of v denotes that the effect of dendritic cells on other modelled cell types should be stronger than that of the keratinocytes. The keratinocyte population would also increase with increase in the parameter n, defining the Hill coefficient characteristic (Fig. 5A). We analyse how other model parameters affect the steadystate response of the keratinocyte for varying levels of IL15 (Fig. 5B, Fig. S3A–C). A steady state analysis of Eq. (14) shows that the population of keratinocytes could increase via a bistable regime (Fig. 5B). It also suggests that the subnetwork would not show bistable behavior if the Hill coefficient \(n < 3\) (Fig. 5B). The exact steady state population of K in the bistable regime could be determined by the initial condition i.e. initial population of Tcells and keratinocytes. As observed with the subnetworks, the high value of K corresponds to a psoriatic state whereas a low value represents a healthy state since the keratinocytes are observed to be hyperproliferative in psoriatic skin^{3,27,28}. These results suggest that an increase in the intracellular IL15 level could lead to a large increase in the keratinocyte population and therefore to a psoriatic state. The increase in the level of IL15 could be caused by several internal and/or external factors.
Figure 5C,D suggest that this bistable phenomena is very robust as the system is bistable in a large region of the two parameter spaces. Examples of dynamics in the monostable and bistable region are shown in Fig. 5E,F. Note that in the bistable region, the system reaches two distinct steady states with the same parameter combination for different initial conditions.
In order to find the source of bistability in this subnetwork, we remove the auto activation of keratinocytes. It leads to the following modification to Eq. (13)
Now, there is only one steady state: \(L_S= \frac{a_L+b_{IL15}D^* \frac{K_{S}^n}{1+K_{S}^n}}{\mu _L k_K}, K_S=\frac{a_K+b_{IL15}D^*}{\mu _K k_K}\).
Here we observe that the subnetwork does not show bistability in the absence of the self activation of keratinocytes. This is not surprising as the presence of a positive feedback loop is a necessary condition for a reaction network to exhibit bistability^{35,36,37}. Therefore, our results suggest that the source of bistability in this subnetwork would be the self activation loop of keratinocytes.
A novel therapy for treating Psoriasis by exploiting bistability
The present study suggests that the subnetworks involving the IL23/IL17 axis and IL15 cytokine could cause psoriasis via bistable regimes, we explore if this behavior can be exploited in order to find a new treatment option. Our approach is demonstrated for the subnetwork formed by IL15.
In the model, an IL15 level corresponds to a particular value of parameter \(b^{'}_{IL15}\). To mimic the treatment of employing an anti IL15 drug, \(b^{'}_{IL15}\) is decreased. Also, we assume that the absorption and dynamics of the drug is faster than the dynamics of this subnetwork, and therefore a rapid drop in \(b^{'}_{IL15}\) is assumed following administration of anti IL15 drugs. This drop is modelled by using an inverse rectangular pulse of the following form
where t is in days. The time window \((t_1, t_2 )\) corresponds to the duration in which the signal S drops completely. The selection of time \(t_1\) depends on the dynamics of the subnetwork (and in turn also on the parameter combination), as the signal should fall only after the steady state is reached. The incorporation of an anti IL15 signal S in the system modifies Eq. (14) as follows
This ODE system (Eq. 3) is solved for a parameter combination in which the subnetwork shows bistability (for example the parameter combination in Fig. 5B). The initial condition is chosen such that, eventually, the dynamics of the subnetwork reaches a steady state where \({\bar{K}}\) is high. One such initial condition (\({\bar{L}}=2\) and \({\bar{K}}=4\)) is shown in Fig. 5E,F. For such an initial condition and parameter combination, the subnetwork would reach the steady state in 20 days (Fig. 5E,F). Therefore, an anti IL15 signal is used after 20 days, which causes IL15 to drop fully between 30 and 60 days (Fig. 6A,C). Our results suggest that the keratinocyte population would start decreasing as soon as the signal S drops (Fig. 6B,D). A comparison between Figs. 5E and 6D suggests that the keratinocyte population would eventually reach a lower steady state consistent with the initial condition \({\bar{L}}=0.2\) and \({\bar{K}}=0.2\). Interestingly, even if the signal bounces back to the previous level after 60 days, the system would remain in the lower steady state which corresponds to a psoriasis free state. These results demonstrate that anti IL15 treatment could be a useful treatment for psoriasis. Indeed, IL15 blockade in a xenograft mouse model has been shown to reduce the severity of psoriasis^{38}.
Discussion
In the present work, the asymptotic behavior of the three subnetworks has been investigated. Based on our numerical analysis, we suggest that increases in the level of cytokines IL15, IL17/IL23 and \(\hbox {TNF}_{\alpha }\) would indeed lead to psoriasis but their routes would be different. In the case of a subnetwork involving only \(TNF_\alpha\), the hyperproliferation of keratinocytes would occur monotonically while in subnetworks involving \(IL15\) and IL23/IL17 it could occur through a bistable regime. In the case of \(\hbox {TNF}_{\alpha }\), psoriasis occurs with gradual increase in keratinocytes, therefore, we believe that an anti\(\hbox {TNF}_{\alpha }\) drug should cure psoriasis gradually. In contrast, in the case of cytokines IL17/IL23 and IL15 the increase in population could happen in an abrupt manner thanks to the bistability. These results are consistent with the observation that IL17 inhibitors show quicker response and better efficacy compared to anti\(\hbox {TNF}_\alpha\) biologics^{21,22}. However, based on this study we cannot attribute these observations to bistable behaviour although this warrants further investigation. In addition, here it is demonstrated that the bistable behavior could be an advantage for developing new treatments for psoriasis. The suggested treatment could be a strategy through which first the severity of the psoriasis could be lessened by appropriately perturbing the system even if the clearance of psoriasis is not achievable. Once the severity is reduced, medicines can be prescribed which would reduce or at least maintain the achieved low severity. Our results also suggests that targeting two or more cytokines at a time could be an effective strategy, which parallels movements to advocate bispecific agents^{39}. Interestingly, dual targets have previously been trialled for psoriasis (IL17 with \(\hbox {TNF}_{\alpha }\))^{40}. Based, on our results we argue that targeting IL15 and IL17/IL23 would be more effective than targeting \(\hbox {TNF}_{\alpha }\).
One of the limitations of this work is that only three subnetworks of the full network described in Ref.^{18} are studied. Another limitation is, here, results are shown only for a small region of multidimensional parameter space. Further, while investigating the treatment of psoriasis via ‘antiIL15’ treatment only one type of signal was assumed to have perturbed the system. To overcome these limitations the model should be expanded upon in a number of ways in the future. Firstly, other important components for psoriasis pathogenesis should be incorporated. These would include key immune cells such as macrophages, neutrophils and natural killer cells. Additionally, future models should investigate other cytokines that are implicated in psoriasis, such as IL22. In our model, for simplicity, IL17 represents all members of its family. Future work should address the role of IL17 subtypes. Model results should also be compared with the available experimental data. In future, by exploring patientspecific models and predicting their response to treatments, one could aim for personalized medicine.
Methods
In^{18}, we synthesised existing knowledge into a schematic of the cytokinemediated feedback network between keratinocytes, Tcells and dendritic cells. This schematic is reproduced in Fig. 1A. In order to mediate a tractable analysis of the dynamics of this network, in the context of treatment for psoriasis, we highlight three subnetworks of interest, mediated by one or two cytokines (IL23/IL17, \(\hbox {TNF}_\alpha\) and IL15) in Fig. 1B–D. The dynamics of these subnetworks is the focus of the current study.
The mathematical modelling framework
The population dynamics of the considered cell types is modelled using ordinary differential equations (ODEs) as proposed in our previous general model for psoriasis pathogenesis^{18}.
where \(a_x\) represents a basal rate of increase in the cell population x due to migration or differentiation factors, which is assumed to be independent of the current population x. The third term (\(\mu _x\)) represents the decay in cell population levels (for example apoptosis independent of modelled cytokines). The second term represents a net birth rate which encompasses the cytokinemediated effects of (1) differentiation/maturation, (2) proliferation and (3) apoptosis. We assume a saturation in the birth rate, dependent upon the cell population (x) and therefore use either of following two forms, to account for enhancing or diminishing effects of cytokines:

(i)
if \(b_s\) increases with an increase in the cell population x
$$\begin{aligned} b_s=b_{x}\frac{x^n}{k_x^n+x^n} \end{aligned}$$ 
(ii)
if \(b_s\) decreases with an increase in the cell population
$$\begin{aligned} b_s=b_{x}\frac{k_x^n}{k_x^n+x^n}. \end{aligned}$$Here, \(b_x\) represents the saturated rate of change due to change in the population x. The parameter \(k_x\) can be interpreted as the fraction of the population x required to achieve half the maximal increase/decrease in x due to the population x. The parameter n is the Hill coefficient, which determines the slope of the function \(b_s\).
In the following sections we introduce specific forms of Eq. (4) for the three subnetworks of interest.
\(\text {TNF}_{\alpha }\) subnetwork model
Figure 1B details the interactions between cells driven by \(\text {TNF}_{\alpha }\). Using the general system (Eq. 4), these interactions are modelled using the following set of ODEs:
where L, D and K denote the current population of T cells, matured dendritic cells and keratinocytes, respectively. The parameter \(b_{TNF _{\alpha }}\) represents the maximum rate of increase in D and K due to the effect of the cytokine \(\hbox {TNF}_{\alpha }\) which is secreted from all three types of cells. The parameter \(k_D\) denotes the population of dendritic cells required for half of the maximum effect D can achieve on K or D. Similarly, \(k_K\) (or \(k_L\)) represents the population of keratinocytes (or Tcells) required for half of the maximum effect K (or L) can have on D or K.
From Eq. (5), it is obvious that the dynamics of the Tcells is independent of the dynamics of the dendritic cells and keratinocytes. Therefore, a quasisteady state scenario for Tcells can be assumed and the ODE system, Eq. (5), will be predominantly governed by the following two dimensional system
where \(L^*=\frac{a_L}{\mu _L}\) represents the steady state population of Tcells.
In order to further reduce the complexity of this and subsequent systems, we nondimensionalise these equations. Assuming that \({\bar{K}}=\frac{K}{k_k}\), \({\bar{L}}=\frac{L^*}{k_L}\) and \({\bar{D}}=\frac{D}{k_D}\), the Eq. (6) can be represented by the dimensionless system
where \(p=\frac{a_D}{k_D}, b_{TNF}^{'}=\frac{b_{TNF_{\alpha }}}{k_D}, q=\frac{a_{K}}{k_K}\), and \(u=\frac{k_D}{k_K}\)
IL23/IL17 subnetwork model
The cytokines IL23 and IL17 are currently used as targets for the treatment of psoriasis^{20,21,22}, and are therefore crucial components of our study. Figure 1A,C demonstrate that these cytokines form a feedback loop between dendritic cells and Tcells, but that there is no direct link to keratinocytes. Rather, the effect on keratinocytes is mediated by a network of connections involving other cytokines. In order to simplify this scenario, we postulate a net effect of the IL23/IL17 axis on the keratinocyte populations. We consider that this net effect can either be activating or inhibiting, which gives rise to the following two cases:
Case (1): The net effect of D and L on K is activating
where \(A=D+L\). L, D and K represent the population of modelled cells as for the previous subnetworks. Also, parameters \(b_{IL23}\), \(b_{IL17}\) and \(b_{net}\) can be defined as for the corresponding terms in the previous subnetworks. The dimensionless form of Eq. (8) becomes
where \({\bar{L}}=\frac{L}{k_L}\), \({\bar{D}}=\frac{D}{k_D}\), \({\bar{K}}=\frac{K}{k_K}\), \({\bar{A}}={\bar{D}}+{\bar{L}}\), \(r= \frac{a_L}{k_L}, b_{IL23}^{'}=\frac{b_{IL23}}{k_L}, b_{IL17}^{'}=\frac{b_{IL17}}{k_D}\) and \(b_{net}^{'}= \frac{b_{net}}{k_K}\). Parameters p and q, are defined same as in the previous subnetwork.
Case (2): The net effect of D and L on K is inhibitory. The population dynamics of the subnetwork is
where \(A=D+L\). Other terms have the same meaning as in case (1). The dimensionless representation is
where \({\bar{L}}=\frac{L}{k_L}\), \({\bar{D}}=\frac{D}{k_D}\), \({\bar{K}}=\frac{K}{k_K}\) and \({\bar{A}}={\bar{D}}+{\bar{L}}\). p, q, r, \(b_{IL23}^{'}\), \(b_{IL17}^{'}\), and \(_{net}^{'}\) are defined same as in Case (1).
IL15 subnetwork model
IL15 arises as an interesting cytokine as it mediates selfactivation of keratinocytes (Fig. 1A). In addition, it promotes the production of cytokines such as IFN\(\gamma\), \(TNF_{\alpha }\) and IL17 and therefore a potentially important mediator of the psoriasis phenotype^{40} . The IL15 subnetwork (Fig. 1D) is modelled as follows:
where L, D and K have the same meaning as in the previous subnetworks. The parameter \(b_{IL15}\) represents the maximum rate of increase in Tcells and keratinocyte population due to changes in the modelled cell types; it reflects the effect of changing the level of IL15. Note that changes in D and K will lead to changes in the level of IL15 as it is secreted from D and K. The parameter \(k_K\) can be described as the population of keratinocytes required for half of the maximum effect K can achieve on K or L. Similarly, \(k_D\) can be defined as the population of dendritic cells required for half of the maximum effect D can have on L or K.
In Eq. (12), the dynamics of the dendritic cells is independent of the dynamics of the Tcells and keratinocytes. Therefore, a quasisteady state scenario is assumed for dendritic cells and the ODE system (Eq. 12) will be largely governed by the following two dimensional system
with \(D^{*}= \frac{\left( \frac{a_{D}}{\mu _{D}}\right) ^n}{k_D^n +\left( \frac{a_{D}}{\mu _{D}}\right) ^n}\). The corresponding dimensionless system is given by
where \({\bar{K}}=\frac{K}{k_K}\) and \({\bar{L}}=\frac{L}{k_L}\) are dimensionless quantities and denote the relative population of keratinocytes and Tcells, respectively (relative to \(k_K\)). The \(b_{IL15}^{'}=\frac{b_{IL15}D^{*}}{k_K}, v=\frac{k_K}{k_L}\). Parameters q and r are defined same as in case of subnetwork involving \(TNF_{\alpha }\).
Model parameters
The parameters of the system, and their default values are given in Table 1. The rate of apoptosis of Tcells (\(\mu _L\)) has been estimated as 0.5 day\(^{1}\) during their response to Lymphocytic Choriomeningitis Virus^{41}. The rate of apoptosis of cutaneous dendritic cells (\(\mu _D\)) has been observed to be approximately 0.2 day\(^{1}\)^{42}. The rate of apoptosis of keratinocytes (\(\mu _K\) ) is unknown therefore given these two estimates, we choose a similar value, 0.5 day\(^{1}\), for that of keratinocytes.
In the subsequent section, we use the dynamics of the \(\hbox {TNF}_\alpha\) subnetwork to determine further parameters, based on the existence of high and low keratinocyte states.
Data availability
The datasets generated for this study can be found in the manuscript and supplementary Data.
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Acknowledgements
RP acknowledges the Department of Science and Technology, India for the support through the DSTINSPIRE Faculty Award (DST/INSPIRE/04/2015/001939). This work was supported by the Engineering and Physical Sciences Research Council (EPSRC), United Kingdom (Grant numbers EP/J018295/1, EP/J018392/1, EP/N014391/1). The contribution of RP was also supported by the later Grant. This work was generously supported by the Welcome Trust Institutional Strategic Support Award (204909/Z/16/Z) too. The contribution of MG was supported by the EPSRC via EP/N014391/1 and a Wellcome Trust Institutional Strategic Support Award (WT105618MA). The contribution of YA was generously supported by the Wellcome Trust Institutional Strategic Support Award (WT105618MA).
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R.P. and M.G. designed the research and R.P. carried out all simulations and analyzed the data. Y.A., R.K.M. and S.K.S. helped in the interpretation of data and writing of the manuscript. All the authors contributed in the preparation of the article.
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The author RKM is currently employed by iOligos Technologies Private Limited, Noida, India. No work occurred in his current (commercial) affiliation. All authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
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Pandey, R., AlNuaimi, Y., Mishra, R.K. et al. Role of subnetworks mediated by \(\hbox {TNF}\alpha\), IL23/IL17 and IL15 in a network involved in the pathogenesis of psoriasis. Sci Rep 11, 2204 (2021). https://doi.org/10.1038/s41598020805077
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DOI: https://doi.org/10.1038/s41598020805077
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