A systematic approach to analyze T cell migration: application to mouse melanoma tumors

A systematic approach to analyze T cell migration: application to mouse melanoma tumors | bioRxiv /* */ /* */ <!-- <!-- /*! * yepnope1.5.4 * (c) WTFPL, GPLv2 */ (function(a,b,c){function d(a){return"[object Function]"==o.call(a)}function e(a){return"string"==typeof a}function f(){}function g(a){return!a||"loaded"==a||"complete"==a||"uninitialized"==a}function h(){var a=p.shift();q=1,a?a.t?m(function(){("c"==a.t?B.injectCss:B.injectJs)(a.s,0,a.a,a.x,a.e,1)},0):(a(),h()):q=0}function i(a,c,d,e,f,i,j){function k(b){if(!o&&g(l.readyState)&&(u.r=o=1,!q&&h(),l.onload=l.onreadystatechange=null,b)){"img"!=a&&m(function(){t.removeChild(l)},50);for(var d in y[c])y[c].hasOwnProperty(d)&&y[c][d].onload()}}var j=j||B.errorTimeout,l=b.createElement(a),o=0,r=0,u={t:d,s:c,e:f,a:i,x:j};1===y[c]&&(r=1,y[c]=[]),"object"==a?l.data=c:(l.src=c,l.type=a),l.width=l.height="0",l.onerror=l.onload=l.onreadystatechange=function(){k.call(this,r)},p.splice(e,0,u),"img"!=a&&(r||2===y[c]?(t.insertBefore(l,s?null:n),m(k,j)):y[c].push(l))}function j(a,b,c,d,f){return q=0,b=b||"j",e(a)?i("c"==b?v:u,a,b,this.i++,c,d,f):(p.splice(this.i++,0,a),1==p.length&&h()),this}function k(){var a=B;return a.loader={load:j,i:0},a}var l=b.documentElement,m=a.setTimeout,n=b.getElementsByTagName("script")[0],o={}.toString,p=[],q=0,r="MozAppearance"in l.style,s=r&&!!b.createRange().compareNode,t=s?l:n.parentNode,l=a.opera&&"[object Opera]"==o.call(a.opera),l=!!b.attachEvent&&!l,u=r?"object":l?"script":"img",v=l?"script":u,w=Array.isArray||function(a){return"[object Array]"==o.call(a)},x=[],y={},z={timeout:function(a,b){return b.length&&(a.timeout=b[0]),a}},A,B;B=function(a){function b(a){var a=a.split("!"),b=x.length,c=a.pop(),d=a.length,c={url:c,origUrl:c,prefixes:a},e,f,g;for(f=0;f<d;f++)g=a[f].split("="),(e=z[g.shift()])&&(c=e(c,g));for(f=0;f<b;f++)c=x[f](c);return c}function g(a,e,f,g,h){var i=b(a),j=i.autoCallback;i.url.split(".").pop().split("?").shift(),i.bypass||(e&&(e=d(e)?e:e[a]||e[g]||e[a.split("/").pop().split("?")[0]]),i.instead?i.instead(a,e,f,g,h):(y[i.url]?i.noexec=!0:y[i.url]=1,f.load(i.url,i.forceCSS||!i.forceJS&&"css"==i.url.split(".").pop().split("?").shift()?"c":c,i.noexec,i.attrs,i.timeout),(d(e)||d(j))&&f.load(function(){k(),e&&e(i.origUrl,h,g),j&&j(i.origUrl,h,g),y[i.url]=2})))}function h(a,b){function c(a,c){if(a){if(e(a))c||(j=function(){var a=[].slice.call(arguments);k.apply(this,a),l()}),g(a,j,b,0,h);else if(Object(a)===a)for(n in m=function(){var b=0,c;for(c in a)a.hasOwnProperty(c)&&b++;return b}(),a)a.hasOwnProperty(n)&&(!c&&!--m&&(d(j)?j=function(){var a=[].slice.call(arguments);k.apply(this,a),l()}:j[n]=function(a){return function(){var b=[].slice.call(arguments);a&&a.apply(this,b),l()}}(k[n])),g(a[n],j,b,n,h))}else!c&&l()}var h=!!a.test,i=a.load||a.both,j=a.callback||f,k=j,l=a.complete||f,m,n;c(h?a.yep:a.nope,!!i),i&&c(i)}var i,j,l=this.yepnope.loader;if(e(a))g(a,0,l,0);else if(w(a))for(i=0;i (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0];var j=d.createElement(s);var dl=l!='dataLayer'?'&l='+l:'';j.src='//www.googletagmanager.com/gtm.js?id='+i+dl;j.type='text/javascript';j.async=true;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-M677548'); Skip to main content Home About Submit ALERTS / RSS Search for this keyword Advanced Search New Results A systematic approach to analyze T cell migration: application to mouse melanoma tumors Nikolaos Memmos , Jason S. Mitchell , Brian T. Fife , David Masopust , View ORCID Profile David J. Odde doi: https://doi.org/10.1101/2025.04.29.651285 Nikolaos Memmos 1 University of Minnesota, Department of Biomedical Engineering , Minneapolis, Minnesota, USA Find this author on Google Scholar Find this author on PubMed Search for this author on this site Jason S. Mitchell 2 University of Minnesota, Center for Immunology , Minneapolis, Minnesota, USA Find this author on Google Scholar Find this author on PubMed Search for this author on this site Brian T. Fife 2 University of Minnesota, Center for Immunology , Minneapolis, Minnesota, USA 3 University of Minnesota, Department of Medicine, Division of Rheumatic and Autoimmune Diseases, Center for Immunology, Center for Autoimmune Disease Research , Minneapolis, Minnesota, USA Find this author on Google Scholar Find this author on PubMed Search for this author on this site David Masopust 2 University of Minnesota, Center for Immunology , Minneapolis, Minnesota, USA 4 University of Minnesota, Department of Microbiology and Immunology , Minneapolis, Minnesota, USA Find this author on Google Scholar Find this author on PubMed Search for this author on this site David J. Odde 1 University of Minnesota, Department of Biomedical Engineering , Minneapolis, Minnesota, USA Find this author on Google Scholar Find this author on PubMed Search for this author on this site ORCID record for David J. Odde For correspondence: oddex002{at}umn.edu Abstract Full Text Info/History Metrics Preview PDF Abstract T cells must assess and choose between surveilling large areas, but also engage efficiently with the target cells. This process is translated into variations in speed and turning angle of T cells. In this study, we propose a generalized algorithm to analyze cell migration data with focus on CD8+ T cells, using clustering technique to identify the number of different migration states and Hidden Markov Model to capture the dynamical switching between them. The algorithm only requires a set of position observations in a series of times, independent of other factors. While this study focuses on CD8+ T cell migration, this approach can potentially be used broadly to study the migration of other cell types as well. For the current analysis, low and high avidity T cells in melanoma tumors were tracked ex vivo using two-photon microscopy. Our findings suggest that CD8+ T cells follow a two-state migration dynamic, with one state being faster, while the other slower and more localized. Moreover, we established a statistical methodology to analyze T cell migration to assess whether there is true variability in cell speeds as distinguished from stochastic fluctuations about a single speed, and it can be applied across different experimental platforms. Introduction Cell migration is a crucial function across different living organisms and scales[ 1 ]. The effectiveness of the immune response depends on how efficiently specific types of immune cells, such as macrophages, Natural Killer (NK) cells, and CD8+ T cells, migrate to locate and eliminate pathogens in the body[ 2 ], [ 3 ], [ 4 ], [ 5 ], [ 6 ]. CD8+ T cells are part of the adaptive immune system and have a key role in immune surveillance for pathogens and tumor cells in the body. Their high specificity resulted in a rapid rise of CD8+ T cell-based cancer immunotherapies[ 7 ], [ 8 ], such as chimeric antigen T cell therapies and tumor-infiltrating lymphocyte therapies. The higher efficacy of T cell immunotherapies for hematological malignancies versus solid tumors, because of the easier tumor accessibility and better T cell survival, suggests the key role of CD8+ T cell migration as a limiting factor regarding the optimization of immunotherapies by tuning their ability to infiltrate and navigate into complex microenvironments for solid tumors[ 9 ], [ 10 ]. CD8+ T cells have been shown to be present in many different solid tumors, such as glioblastoma, melanoma, and lung squamous cell carcinoma[ 11 ], [ 12 ], [ 13 ]. Lymphocyte migration disruption, for instance, can be a potential target for anti-inflammatory drugs for diseases such as Crohn’s disease, multiple sclerosis, and asthma[ 14 ], [ 15 ], [ 16 ], [ 17 ], [ 18 ], [ 19 ]. Thus, understanding CD8+ T cell migration is crucial for optimizing the efficacy of immunotherapies and controlling the immune response generally. Many statistical models have been proposed to describe cell motion during migration. The simplest is the Brownian motion model, which was named after botanist Robert Brown who first observed that the motion of pollen into the water follows this pattern in 1827[ 20 ]. Later in 1855, Adolf Fick recognized that diffusion is analogous to heat transfer, so he proposed the diffusion equation to describe the transport of nutrients through membranes[ 21 ]. However, it was not until 1905 that Albert Einstein derived the diffusion equation in terms of probability density and postulated it[ 20 ], [ 22 ]. The three postulates are: i) two consecutive time intervals are independent and as a result uncorrelated, ii) displacements are identically distributed during each time interval and iii) the second moment of the displacements exist[ 23 ]. Brownian motion is characterized by a linear mean-squared displacement (MSD) with respect to time interval over which a displacement occurs. While this model is widely used in biology to describe the motion of cells, in reality, cell migration often violates the first postulate of the Brownian motion model and other models have been proposed, such as the persistent random walk (PRW) model, which parameterizes cell motility in terms of speed, S , and persistence time, P [ 24 ]. The PRW model is analogous to the Ornstein-Uhlenbeck stochastic process and has been used extensively to model cell migration when cells exhibit correlations in displacement, while the Brownian motion model is a subcase of the PRW model under the assumption that the system is overdamped[ 25 ] or when the sampling time interval is longer than the persistence time in the case of experimental data. It is characterized by a MSD proportional to the square of time interval at short times, and linear for long times. However, cell migration strongly depends on the properties of the extracellular environment and often deviates from the Brownian motion and PRW models. Wu et al. showed that cells can exhibit anisotropic behavior in 3D environments, compared to 2D environments[ 26 ]. Moreover, anomalous diffusion models have been proposed to explain experimental data that deviate from the linear relationship between MSD and time. Dieterich et al. studied the migration of NHE + and NHE-deficient cells from the renal epithelial Madin– Darby canine kidney cell strains and concluded that both populations follow a superdiffusive migration pattern[ 27 ]. CD8+ T cells are a characteristic example where migration cannot be described by the traditional Brownian motion and PRW mathematical models. Boisfleury-Chevance et al. showed that lymphocytes exhibit a completely different migratory behavior compared to granulocytes and monocytes and speculated that lymphocytes follow a two-state migration pattern, while there is no closed-form mathematical model to describe their motility patterns, unless in the case of long times[ 28 ]. Sadjadi et al. also studied the migration of T cells in 3D collagen matrices and pointed out the existence of three motility types: slow, fast and mixed[ 29 ]. They proposed that slow moving T cells create channels for fast T cells in 3D matrices and their motion can be described by a two-state random walk. In both papers, however, the number of substates and the distributions of the observables (i.e. step length and turning angle) on each substate are not well- established. Many different models have been proposed to explain CD8+ T cell migratory behavior. For example, Beauchemin et al. proposed a model that is described by three parameters, the mean free time, tfree , a constant speed vfree , and a pause time tpause [ 30 ]. During the mean free time, the cell moves in a straight line with constant speed and pauses until it will pick a new random direction in which to move. However, the mean-squared displacement is sensitive to the number of data points, the signal-to-noise ratio, and the correlation between the MSD values, which can lead to incorrect estimations [ 31 ], [ 32 ], [ 33 ], [ 34 ]. A plethora of the proposed models have been initially used in animal ecology to explain the different motion patterns in animal movement. Viswanathan et al. first proposed Lévy flight as model that can explain the searching process of animals for targets[ 35 ]. Lévy walk and Lévy flight are a class of random walk models where their step length, λ , distribution is characterized by a power-law relationship, P ( λ )∼ λ −(1+ μ ) , where μ is the power-low exponent and belongs to (0, 2][ 36 ]. These two terms are often used interchangeably in literature; however, they are distinct. Lévy flight assumes that the displacement between two points is discontinuous, compared to Lévy walk[ 37 ]. The argument for using Lévy walk over other models is that it optimizes the searching strategies for living organisms, when the targets (e.g., food) are sparse[ 38 ]. This model has also been used to explain the behavior of marine predators, bacteria, and human migration[ 39 ], [ 40 ], [ 41 ], [ 42 ]. Harris et al. studied the CD8+ T cell migration in mouse brains infected with the pathogen toxoplasma gondii [ 43 ]. They demonstrated that the chemokine CXCL10 enhances the ability of CD8+ T cells to efficiently identify the target cells, resulting in shorter capture times, and CD8+ T cells exhibit a generalized Lévy walk behavior. However, a lot of criticism arose about the Lévy walk model regarding both its biological and statistical grounding[ 44 ], [ 45 ], [ 46 ]. In particular, Lévy walk assumes that displacements are uncorrelated, while the step length, drawn from a heavy-tail distribution with infinite variance, can result in rare, unrealistically large displacements, even though this limitation can be overcome to some degree with truncated distributions. Furthermore, Edwards et al. also reported that the likelihood methods used to infer Lévy walk are inaccurate[ 45 ]. Banigan et al. proposed a Gaussian Mixture Model (GMM) to describe the non- Brownian behavior of CD8+ T cells in uninflamed lymph nodes[ 47 ]. While this model captures the existence of subpopulations based on their migration features, it fails to capture the possibility of the dynamic transitions between these populations. Jerison et al. showed that T cells in live larval zebrafish follow a two-state migration manner and proposed a variation of the PRW model, where persistence time, P , is linearly dependent on speed, S , which they called a speed-persistence coupling model[ 48 ]. While this speed-persistence coupling model is true in this system, it is not clear that it applies to other systems. Hidden Markov Models (HMMs) have been widely used to describe the relationship between a sequence of observed data (observables) and events that are not observed directly (hidden events). HMMs were originally used in speech recognition but also gained popularity in studying biological sequences (e.g., DNA sequencing)[ 49 ], [ 50 ]. In cell migration, HMMs have been used to annotate and infer the dynamical switching between distinct morphological states, but also to describe the stochastic behavior of cell movement[ 51 ], [ 52 ], [ 53 ], [ 54 ]. For example, Degerman et al. applied a two-state HMM to glial progenitor cells and type-2 astrocytes and showed that the latter follow a directed movement pattern 2/3 of the time, compared to the first that behaved mostly as random walkers[ 51 ]. Torkashvand et al. proposed to initially cluster CD8+ T cells based on their motility pattern (Lévy and Brownian types) and the length of their step increments (short and long increments) to capture the heterogeneity across the cells[ 55 ]. This analysis resulted in 4 different clusters and inside each cluster homogeneous and non- homogeneous HMMs with 2 and 3 hidden states to capture the heterogeneity within the cell tracks. While in the current literature there is ample speculation of the existence of substates in T cell migration, there is no algorithm that rigorously establishes the minimum number of substates. In this study, we propose a generalized algorithm to test for heterogeneity in cell migration and the possibility of dynamic transition between different speed states with application to CD8+ T cells ex vivo in a mouse melanoma model. We utilized published 3D two- photon microscopy data from melanoma tumors that include high and low avidity CD8+ T cells[ 56 ]. Avidity is defined as the total binding strength of the receptors of CD8+ T cells with the target cells. Our analysis shows CD8+ T cells, regardless of avidity type, move in both slow- and fast-moving states and switch dynamically between them. Materials and Methods T cell migration in mouse melanoma tumors data Four datasets from B16F10 melanoma tumors from transgenic mice were obtained via ex vivo two-photon microscopy, which were previously published by Tucker et al. [ 56 ]. The transgenic mice produced both low and high avidity CD8+ T cells which were specifically recognizing the tumor Ag tyrosinase-related protein 2 (TRP2). The CD8+ T cells were purified from the spleen and lymph node and then stimulated in vitro with interleukin-12 (IL-12) to prevent exhaustion. Subsequently, the CD8+ T cells were transferred intravenously to the mice. The low and high avidity CD8+ T cells were identified with Brainbow tdTomato and UBC-GFP respectively and tracked in 3D with Imaris software with a frame rate equal to 32 seconds/frame. To reduce bias, only tracks with consecutive time points were allowed in the analysis, otherwise they were considered as separate tracks (e.g. so there were no missing points within one track). In total, N=271 low and N=266 high avidity CD8+ T cells, with 5204 and 5067 total time points respectively, were analyzed. Because there was potentially a significant difference between the migration pattern of the two populations, low and high avidity CD8+ T cell populations were analyzed separately. The total imaging time and time window were similar for the high- and low- avidity datasets ( Table S1 ). Statistical analysis The time averaged mean-squared displacement (TAMSD) and the displacement autocorrelation function were calculated for each individual cell. Assuming a time series of 3D positions r 1 , r 2 , …, r N with N data points k equally spaced time lags, where each vector ri , corresponds to the position of the cell, ri = (xi, yi, zi) , at time point i , the MSD was estimated using the overlapping sampling method[ 57 ]: Displacements were calculated with first differencing from the position datasets to obtain detrended time series. Thus, it was assumed that the time series of the displacements have a mean equal to zero, which is valid for detrended datasets[ 58 ], where the autocorrelation function for each individual cell is given by: To assess autocorrelation function at the population level, the weighted average ACF based on the number of data points for each cell was estimated. The 95% confidence interval was calculated based on Equation S1. Gaussian Mixture Model The Gaussian Mixture Model (GMM) was applied to the distribution of the displacements of the total number of CD8+ T cells and all time points to identify the number of migration states[ 47 ]. In this clustering method, the distribution is approximated as a linear superposition of M Gaussian distributions: where Δr are the displacements, φk are the mixing coefficients and must obey the following conditions: and For the purposes of this study, it was assumed that the mean of the displacement Gaussians, μk , is zero and motion is isotropic, so that equation 3 can be written as: To identify the set of parameters { φm , σ 2 m } which maximize the likelihood of the distribution of the displacements ( equation 7 ), the Expectation-Maximization algorithm has been used[ 59 ]. The Bayesian Information Criterion (BIC) and Akaike Information Criterion (AIC) along with the Elbow Method were used to decide the number of states[ 60 ]. Hidden Markov Model To infer the parameters of possible dynamic switching between states, a Hidden Markov Model (HMM) was applied, using the momentuHMM package, in combination with modified scripts from the moveHMM package, in R language[ 61 ], [ 62 ]. The number of hidden states was dictated by the clustering analysis with GMM as described previously[ 47 ]. The observable variables of the model are { s, θ, Δz }, where s is the step length of a cell projected into the xy- plane, θ is the turning angle projected onto the xy plane and Δz is the displacement in the direction of the z-axis. To fit the HMM, it was assumed that s , θ and Δz follow a gamma distribution, von Mises distribution, and logistic distribution, respectively: where α and λs are the shape and scale of Gamma distribution respectively, μθ and κ are the location and the concentration parameters of the Von Mises distribution, and μΔz and λΔz are the location and scale parameters of the logistic distribution, respectively. The caveat of the gamma distribution is that it cannot model steps with zero length, thus a zero-inflated distribution is needed, which assigns a probability z to observe a zero value and a 1-z probability to observe a positive value based on a gamma distribution[ 61 ], [ 63 ]. It was assumed that the von Mises distribution is symmetric around zero, so μ θ = 0. The turning angle, θ, was calculated using the atan2 function to constrain it in the (- π , π ] range: where x, y are the xy-coordinates, indices 1, 2, and 3 correspond to 3 consecutive time points in a cell track. To estimate the parameters, a numerical maximization of the likelihood was employed[ 64 ], [ 65 ]. Results T cell migration heterogeneity Infiltrated CD8+ T cells in B16 melanoma tumors were tracked using ImageJ for a period of 30 minutes at a frame time interval of 32 seconds. The red cells correspond to low avidity T cells, while the green ones correspond to high avidity T cells ( Figure 1A ). Compared to other types of immune cells, like monocytes and granulocytes, both T cell populations follow a distinctive migration pattern that does not appear to be described by a single migration speed, but instead appears to migrate in a bimodal pattern, consisting of distinct slow- and fast-moving states ( Figure 1B-C )[ 28 ]. The slow-moving state tends to be more localized, possibly because of the engagement of T cells with target cells (e.g. dendritic and tumor cells). On the other hand, the fast-moving state is persistent in a certain direction. Download figure Open in new tab Figure 1. A ) Example of a mouse melanoma migration dataset where green cells correspond to high avidity CD8+ T cells, red to low avidity CD8+ T cells, blue channel on collagen and yellow on dendritic cells. B ) Examples of fast/persistent (left) and slow/localized (right) migration patterns are evident for both low and high avidity T cells. C-D ) An xy-projection of ( C ) low and ( D ) high avidity CD8+ T cell migratory tracks. It appears by eye that the cells migrate in either a localized (slow) or persistent (fast) manner. . The distribution of the displacements of the low and high avidity CD8+ T cells across x , y and z were compared using the Kolmogorov-Smirnov (KS) test to decide if the two populations should be analyzed together. The KS test revealed that the two populations differ, with p-values equal to 0.024, 0.0001 and 0.016 for x , y and z axes respectively, which may be attributed to the different dissociation rates ( Figure S1 ). To assess whether there is variability with respect to migration pattern among individual T cells, the time averaged mean-squared displacement for each individual cell (TAMSD, Figure 2A ) was calculated based on equation (1) . The average displacement autocorrelation function for the T cell populations was also estimated, based on equation (2) , to evaluate whether or not T cells exhibit directional memory (ACF, Figure 2B-C ). The TAMSD is a first estimator of the variability in T cell motility behavior, while the average autocorrelation function describes the directionality and persistence of the T cell migration. Figure 2A shows that TAMSD can vary largely between CD8+ T cells, where low values suggest that T cells perform a localized searching because of their interactions with target cells, while increasing values correspond to a persistent random walk. However, such variations on a cell-to-cell basis can also result from a single random walk model applying to all cells due to stochastic fluctuations. On a population level, the average autocorrelation function was calculated and fitted to a simple exponential function, e − t / τ , where τ is the decay time constant. For the low avidity T cells, the decay time constant is τ low = 0.690 ± 0.035 minutes and for the high avidity, τ high = 0.632 ± 0.030 minutes, again confirming that migration differences between high- and low-affinity T cells are modest. The low avidity T cells exhibit a slightly higher decay time constant, most likely because they spend more time in a persistent migration pattern without efficiently associating with target cells. Both numbers are close to what has been previously observed experimentally for lymphocyte migration in vitro [ 28 ]. Download figure Open in new tab Figure 2. ( A ) Selected examples of the time averaged mean-squared displacement (TAMSD) of individual CD8+ T cells and ( B-C ) the average autocorrelation function over all high and low avidity T cells respectively. The average time window between the time points is 32.339 sec. The black points are the experimental data, the blue curve the fitted exponential function, e − t / τ , and the red dashed line corresponding to the 95% confidence interval. The τ low is equal to 0.6901 ± 0.0348 minutes and the τ high to 0.6319 ± 0.0301 minutes. The normality tests reject the possibility of a single Gaussian In the case of Brownian motion, the distribution of displacements follows a Gaussian distribution: where Δr is the displacement of T cells, n is the number of dimensions, D is the diffusion (or random motility) coefficient, and Δt is the time increment. To evaluate if the T cells obeyed a Brownian motion pattern at the population level, we performed normality tests based on the total distribution of the displacements ( Figure 3A , C ) for each axis, using a QQ-plot ( Figure 3B , D ) and Anderson-Darling test ( Table S3 ). Based on the QQ-plots, it is shown that the data (red points) diverge significantly from the Gaussian distribution (black line) across every direction and for both low and high avidity T cells. The deviation in both the positive and negative tails of the distribution suggests that the distributions are heavy tailed relative to a Gaussian. The same conclusion is drawn from the Anderson-Darling normality test, which rejects the null hypothesis of normality. Overall, we conclude that T cell migration is not well described by a random walk with q single random motility coefficient. Download figure Open in new tab Figure 3. ( A, C ) The distribution of the displacements for low avidity CD8+ T cells and all time points across each axis for low and high avidity T cells respectively. ( B, D ) The corresponding QQ-plots for each distribution. The black line corresponds to theoretical expectation for a normal distribution and red points correspond to the data. The deviations at the distribution extremes implies that displacements are not well-described by a Gaussian distribution, but rather are more heavy-tailed. The Gaussian Mixture Model indicates a two-state migration Since the normality tests rejected the possibility of a universal migration pattern described as Brownian motion, the question remains if a multiple state model can be used to model the migration of T cells. The existence of a heavy tailed displacement distribution suggests that T cell migration can be described by a Lévy walk[ 43 ]. However, this model has been criticized for its ability to describe foraging processes, because of the infinite variances and the inappropriate use of likelihood function [ 44 ], [ 45 ], [ 46 ]. Previously, Banigan et al . used a Gaussian Mixture Model to describe T cell migration in the lymph node in the absence of inflammation[ 47 ]. In our case, we used the same approach to reveal how many clusters of migration states exist in both T cell populations, but may not be a completely adequate model, since it does not capture the potential for transition between the faster and slower states. The relatively weak correlation in the displacements ( Figure 2B-C ), shows the weak persistence in the population migration behavior and justifies the use of GMM as a clustering method. To perform the clustering, it was assumed that the mean of the distributions is equal to zero and the only free parameters were the weights and the covariance matrices for each component. Moreover, the assumption of isotropic migration of the CD8+ T cells in the tumors results in less free parameters, since the covariance can be described with a single parameter and takes the form σ 2 I , where I is the identity matrix. The tolerance for convergence of the maximum likelihood was set to 10 -6 ( Figure S2 ). Both Akaike and Bayesian information criteria indicated that the optimal number of clusters to describe the probabilities was equal to two based on the elbow method ( Figure 4 ). Higher numbers of states do not reduce the AIC and BIC further, indicating that two speeds (one fast, one slow) is the minimum number of substates. Our result of two substates agrees with Banigan et al ., who also identified two T cell speeds[ 47 ]. From Tables 1 and 2 , it is shown that a similar percentage of low and high avidity T cells belong to each state, however low avidity T cells appear to have larger variances on each state, because of their slightly higher speed, compared to high avidity T cells. While helpful in testing whether the data can be explained by a single migration model or whether substates exist, GMM does not capture potential dynamic switching between the states and should be used only as a clustering method when appropriate to determine the number of distinct migration speeds justified by the data. Download figure Open in new tab Figure 4. ( A-B ) The AIC and BIC scores with respect to number of components of the Gaussian Mixture Model for low and ( C-D ) high avidity CD8+ T cells. Both criteria indicate that the distribution of the displacements is better described with a two-state model, based on the elbow method. The Hidden Markov Model captures the dynamic behavior of T cells The GMM indicated the existence of a two-state migratory behavior of T cells (i.e. a fast state and a slow state), however it cannot capture their dynamic behavior. For that reason, a Hidden Markov Model was applied to identify the transition probabilities between the fast and slow states. The observables for the HMM were the step length of the T cells in the xy -plane, the turning angle and the displacement in the z -axis. The probability distributions of these observables are described by the equations ( 8 )-( 10 ). In Figure 5 it is shown that the step length distribution in the fast-moving state is broader (i.e. is faster, as expected), while the angle distribution is concentrated around zero (i.e. is more persistent, tending not to turn as much). Download figure Open in new tab Figure 5. ( A ) The step distribution, ( B ) the turning angle distribution and ( C ) the distribution of the displacements across the z-axis of high avidity T cells. The red and blue lines correspond to the fast- and slow-moving states respectively. The dark red curves correspond to the slow-moving state, the blue to the fast-moving state and the black to the total density. This observation confirms our initial speculation, that persistent motion correlates with faster migration. On the other hand, slow migration is accompanied by an almost uniform turning angle distribution, consistent with a random exploration of a small region in space. Furthermore, low avidity T cells move somewhat faster than high avidity T cells, in both their fast state and their slow state (Supplementary Information). Tables 3 and 4 show the transition probabilities between the states for low and high avidity T cells, respectively. Each row corresponds to the current state and each column to the potential next state, while the sum of each row’s elements is equal to one (i.e. they must either remain in their current state or switch to the other state). We find that the average probability for a T cell to change state is equal to 0.136, which can be translated to a transition rate of 0.25 min -1 approximately (or equivalently switching every ∼4 min). Thus, the probability of T cells changing state, in both directions, in one minute is around 25%, which is independent of their avidity. The fitted models were evaluated using their residuals (Supplementary Figure S3-S4 ). The QQ-plots of the pseudo-residuals, also known as quantile residuals, show that they are in good agreement with a normal distribution, while the ACF of the pseudo-residuals is weak for every observable, suggesting a good fit of the HMM to the data. In the case of high avidity T cells, the ACF of the pseudo-residuals for the step length, there is a relatively higher autocorrelation which can be attributed to their stronger interactions with the target cells or other environmental variables. However, in both cases, we consider that the HMMs satisfy the goodness-of-fit test, using the pseudo-residuals[ 65 ]. The Hidden Markov Models capture the behavior of the original dataset To validate the fitted Hidden Markov Model for each T cell population, a simulation of 100 T cells for 500 time points, using the fitted HMMs, was performed and the average ACF function was calculated ( Figure 6 ). The time window for the simulations was assumed to be equal to the experimental dataset. Then a single exponential function was fitted as in the experimental data ( Figure 3 ). As it is shown in Figure 6 , the exponential function, e − t / τ , perfectly fits the simulated data. For the low avidity simulated T cells, the time decay constant is 0.701 ± 0.008 minutes and 0.587 ± 0.006 for the high avidity cells. In the case of low avidity T cells, the fitted time decay constant of the simulated data is closer to experimental one ( τ low = 0.6901 minutes), compared to the high avidity T cells ( τ high = 0.632 minutes). This discrepancy is a consequence of the higher autocorrelation in the step length for the high avidity model ( Figure S4 ). In both cases, the average ACFs of the simulated data agree with the experimental ones. Moreover, the fitted HMMs are capturing the same behavior as in the experimental data, with respect to time averaged mean-squared displacement ( Figure S5 ). Download figure Open in new tab Figure 6. Autocorrelation Function of displacements shows only weak and short time-scale positive correlations. The fitted ACF on the simulation data for (A) low and (B) high avidity CD8+ T cells. The blue line corresponds to the fitted curve, the black points on the calculated points and the red line on the significance threshold. For each simulation, 100 cells were used and 500 time points. The time window was assumed to be equal to 32.339 seconds, as in the experimental dataset. Discussion CD8+ T cells play a key role in surveilling the human body and eliminating pathogens and cancer cells. The rapid development of T cell-based cancer immunotherapies dictates the necessity for better engineering of CD8+ T cells to efficiently explore the tumor microenvironment and kill cancer cells. This can be achieved by properly tuning the migration of T cells in tumor microenvironments. Thus, the understanding of the nature of the movement of T cells, the existence of possible substates and their statistical properties is crucial for cancer immunotherapy optimization. In this study, two T cell populations with different avidities were analyzed with respect to their migratory behavior in B16 melanoma tumors, using ex vivo 3D two-photon microscopy imaging [ 56 ]. Both T cell populations displayed two-state migration, with one state being slow and resembling a random walk, while the other being fast and persistent. While these findings have also been observed in previous studies, here we propose a systematic algorithm and approach for analyzing T cell migration data to rigorously identify the number of migration substates, their properties, and the rates of transition among them. Because of the complexity of T cell migration and their importance, numerous mathematical models have been proposed to describe their behavior. It has been shown that naïve T cells follow a diffusive or subdiffusive random walk across different studies[ 66 ], [ 67 ], [ 68 ], [ 69 ]. On the other hand, when T cells are activated, they switch to more complex migration dynamics, which includes both ballistic-type fast migration (with long-tailed displacement distributions) and more localized (slow) search [ 69 ], [ 70 ]. This observation led to the use of the Lévy walk model to describe T cell migration in brain tissue, in response to infection[ 43 ]. While this model seems to explain the behavior of T cells, it fails to capture the autocorrelation between the displacements, which contradicts what happens in nature and based on our analysis, and it has been criticized based on the inappropriate use of likelihood function [ 45 ], [ 46 ], [ 71 ]. Edwards et al . demonstrated that previous studies used the residuals of the regression to estimate the likelihood function of the proposed models, instead of the likelihood of the actual probability distribution, which lead to misleading information criteria[ 45 ]. Furthermore, its validity has not been verified in other tissues[ 69 ]. Banigan et al . has shown that T cells do not follow a Lévy walk in a brain infection model, but a combination of slow- and fast-moving states, using a Gaussian Mixture Model, while Jerison et al. used the speed-persistence coupling model to describe this behavior[ 47 ], [ 48 ]. The findings of these papers agree with what we observe in our case with melanoma- infiltrating CD8+ T cells, but their model is not able to capture the switching between the states, which we now capture with the Hidden Markov Modeling approach. Moreover, the Hidden Markov Model can be applied across different conditions and platforms, compared to a closed- form equation. The similarity of the fast-slow substates suggests that T cell migration mechanics are conserved across infections and tumors. In our analysis, we combined GMM and HMM, to analyze T cell migration in solid tumors, while trying to simultaneously minimize the number of assumptions. Our analysis indicated that CD8+ T cells, in both cases, do not obey pure Brownian motion based on normality tests, while the coexistence of two states was verified with a Gaussian Mixture Model, as in Banigan et al .[ 47 ]. The use of this clustering methodology is justified by the weak displacement autocorrelations. It was observed that low avidity T cells have a longer memory, compared to the high avidity T cells, potentially because of their weaker engagement with the target cells. The clustering analysis identified the number of hidden states that was used in the Hidden Markov Model to model the dynamic switching between the states. Finally, the fitted models were judged based on their ability to capture the behavior of the original dataset, using as metrics primarily the ACF and secondly, the reproduction of the TAMSD curves. The understanding of CD8+ T cell migration can assist in the design of better immunotherapies for cancer treatment. Our analysis rigorously showed that T cells migrate in a two-state model, a slow-moving state with less directional persistence and a fast-moving state with high persistence, and that single cells can switch between these states. T cells can be engineered accordingly to infiltrate the tumor microenvironment and engage with the target cells, which surveil the area effectively. The exact nature is likely dependent on the type of tumor, for example solid tumors may require T cells to remain more in the slow-moving state after infiltration, because of the high density of target cells. Even though the algorithm was applied to CD8+ T cells, the same approach can be applied to other cell types. The two major caveats of our algorithm are the use of the GMM which can only be applied in the case with low autocorrelation with regards to the displacement of the cells. If higher autocorrelations exist, other clustering methods, such as Dynamic Time Wrapping (DTW) should be used to identify the number of the states[ 72 ]. The second caveat is that it does not account for the influence of external factors, such as the effect of other types of cells, and the spatial component. Overall, our method can capture the features of T cell migration in tumor tissue ex vivo , without relying on closed-form equation models and with a small number of assumptions. Conflict of Interest All authors declare that they have no conflicts of interest. Data availability statement The data that support the findings of this study are available from the corresponding author upon reasonable request. Ethics approval statement Experiments were conducted under specific pathogen-free conditions and performed in compliance with relevant laws and guidelines, with approval of the Institutional Animal Care and Use Committee at the University of Minnesota. View this table: View inline View popup Download powerpoint Table 1. The weights ( φ ) and the variances ( σ 2 ) of the two Gaussian mixture components for low avidity T cells. View this table: View inline View popup Download powerpoint Table 2. The weights (φ) and the variances (σ 2 ) of the two Gaussian mixture components for high avidity T cells. View this table: View inline View popup Download powerpoint Table 3. The transition rates of low avidity T cells given in min -1 , based on the r = p / Δt equation, where r is the transition rate, p the probability, and Δt the experimental time window equal to 32.339 sec. The numbers within the parenthesis correspond to the probability values. View this table: View inline View popup Download powerpoint Table 4. The transition rates of high avidity T cells given in min -1 , based on the r = p / Δt equation, where r is the transition rate, p the probability, and Δt the experimental time window equal to 32.339 sec. The numbers within the parenthesis correspond to the probability values. Table of Contents Download figure Open in new tab The results show that a minimum of two migration speeds can be rigorously identified from the data with cells switching between a fast, persistent migratory state, and a slow, random migration state. These results will help in identifying genetic factors that influence rapid migration, among other applications, such as quality control for CAR-T cell migration. Supplementary Information To calculate the 95% confidence interval for white noise autocorrelation function, equation S1 was used where n is the number of time points[1]. To assess the goodness-of-fitness of the HMM model, the normal pseudo-residuals were used[65], [73]. Assume a set of observations x 1 , x 2 , …, x T from a model X t ∼ F t , where F t is the distribution of random variable X t . The normal pseudo-residuals are defined as: where ϕ is the standard normal distribution and F Xt ( x t ) = Pr ( X t ≤ x t ) is the cumulative distribution function or in other words, x t is transformed to z t . If the fitted model is adequate, then z t follows a standard normal distribution. Download figure Open in new tab Figure S1. The displacement ECDF of low and high avidity CD8+ T cells on each axis. The p- values are 0.024, 0.0001 and 0.016 for x, y and z axes respectively with a p-value threshold of 0.05, using Kolmogorov-Smirnov test. The table compares the means and the standard deviations of the distributions of the displacements for low and high avidity T cells. High avidity T cells exhibit a lower standard deviation, and it can be attributed to the weaker dissociation rates. Download figure Open in new tab Figure S2. ( A ) Log-likelihood values with respect to the number of iterations till convergence of GMM for high and ( B ) low avidity CD8+ T cells. Download figure Open in new tab Figure S3. Evaluation of the goodness-of-fit of HMM with pseudo-residuals. ( A , D , G ) The time series, ( B , E , H ) the QQ-plots and ( C , F , I ) the autocorrelation function of the pseudo-residuals for step length, turning angle and Δz respectively for low avidity CD8+ T cells. Deviation from the normal distribution in QQ-plots indicates poor fitting, while high autocorrelation indicates that the model does not capture important factors that influence the migration. Download figure Open in new tab Figure S4. Evaluation of the goodness-of-fit of HMM with pseudo-residuals. ( A , D , G ) The time series, ( B , E , H ) the QQ-plots and ( C , F , I ) the autocorrelation function of the pseudo-residuals for step length, turning angle and Δz respectively for high avidity CD8+ T cells. Deviation from the normal distribution in QQ-plots indicates poor fitting, while high autocorrelation indicates that the model does not capture important factors that influence the migration. Download figure Open in new tab Figure S5. ( A ) Low and ( B ) high avidity time-averaged mean squared displacement plots based on produced simulated data by the Hidden Markov Models. Download figure Open in new tab Figure S6. Example of a T cell trajectory and its dynamic switch between the two states. View this table: View inline View popup Download powerpoint Table S1. Details of the four datasets. View this table: View inline View popup Download powerpoint Table S2. Kolmogorov-Smirnov Test p-values for comparing low and high avidity CD8+ T cell distribution of the displacements. View this table: View inline View popup Download powerpoint Table S3. Values of the Anderson-Darling Normality Test for the whole CD8+ T cell population, low and high avidity CD8+ T cells separated. View this table: View inline View popup Download powerpoint Table S4. Step distribution parameters for the low avidity T cells in μm . View this table: View inline View popup Download powerpoint Table S5. Angle distribution parameters for the low avidity T cells in radians . View this table: View inline View popup Download powerpoint Table S6. Δz distribution parameters for the low avidity T cells in μm . View this table: View inline View popup Download powerpoint Table S7. Initial distribution for the low avidity T cells. View this table: View inline View popup Download powerpoint Table S8. Step distribution parameters for the high avidity T cells in μm . View this table: View inline View popup Download powerpoint Table S9. Angle distribution parameters for the high avidity T cells in radians . View this table: View inline View popup Download powerpoint Table S10. Δz distribution parameters for the high avidity T cells in μm . View this table: View inline View popup Download powerpoint Table S11. Initial distribution for the high avidity T cells. Acknowledgements This work was supported by grants from the National Institutes of Health (grant numbers R01 AI156276 to B.T.F., U54CA210190, U54CA268069, and P01CA254849 to D.J.O.). Multiphoton imaging on the Leica Stellaris 8 DIVE system was made possible by a NIH S10 grant 1S10OD034456-01 (B.T.F.) to the University of Minnesota Center for Immunology. Funder Information Declared National Institutes of Health, https://ror.org/01cwqze88 , R01 AI156276 , U54CA210190 , U54CA268069 , P01CA254849 , 1S10OD034456-01 References [1]. ↵ X. Trepat , Z. Chen , and K. Jacobson , “ Cell Migration ,” Compr Physiol , vol. 2 , no. 4 , pp. 2369 – 2392 , 2012 , doi: 10.1002/CPHY.C110012 . OpenUrl CrossRef PubMed [2]. ↵ W. J. 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Share A systematic approach to analyze T cell migration: application to mouse melanoma tumors Nikolaos Memmos , Jason S. Mitchell , Brian T. Fife , David Masopust , David J. Odde bioRxiv 2025.04.29.651285; doi: https://doi.org/10.1101/2025.04.29.651285 Share This Article: Copy Citation Tools A systematic approach to analyze T cell migration: application to mouse melanoma tumors Nikolaos Memmos , Jason S. Mitchell , Brian T. Fife , David Masopust , David J. 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