On the organization of human T-cell receptor loci: log-periodic distribution of T-cell receptor gene segments

Abstract

The human T-cell repertoire is complex and is generated by the rearrangement of variable (V), diversity (D) and joining (J) segments on the T-cell receptor (TCR) loci. The T-cell repertoire demonstrates self-similarity in terms clonal frequencies when defined by V, D and J gene segment usage; therefore to determine whether the structural ordering of these gene segments on the TCR loci contributes to the observed clonal frequencies, the TCR loci were examined for self-similarity and periodicity in terms of gene segment organization. Logarithmic transformation of numeric sequence order demonstrated that the V and J gene segments for both T-cell receptor α (TRA) and β (TRB) loci are arranged in a self-similar manner when the spacing between the adjacent segments was considered as a function of the size of the neighbouring gene segment, with an average fractal dimension of approximately 1.5. Accounting for the gene segments occurring on helical DNA molecules with a logarithmic distribution, sine and cosine functions of the log-transformed angular coordinates of the start and stop nucleotides of successive TCR gene segments showed an ordered progression from the 5′ to the 3′ end of the locus, supporting a log-periodic organization. T-cell clonal frequency estimates, based on V and J segment usage, from normal stem cell donors were plotted against the V and J segment on TRB locus and demonstrated a periodic distribution. We hypothesize that this quasi-periodic variation in gene-segment representation in the T-cell clonal repertoire may be influenced by the location of the gene segments on the periodic-logarithmically scaled TCR loci. Interactions between the two strands of DNA in the double helix may influence the probability of gene segment usage by means of either constructive or destructive interference resulting from the superposition of the two helices.

1. Introduction

T cells are central to the normal execution of adaptive immunity, allowing identification of a multitude of pathogens and transformed cells encountered in an organism's lifetime. T cells accomplish this task by recognizing peptide–major histocompatibility complex (MHC) complexes by means of hetero-dimeric T-cell receptors (TCRs) expressed on their surface. The TCR serve the primary antigen recognition function in adaptive immune responses. TCRs comprised either an alpha and a beta chain (TCR αβ) in the majority of T cells, or less frequently, gamma and delta chains (TCR γδ). [1] The ability of the human T cells to recognize a vast array of pathogens and initiate specific adaptive immune responses depends on the diversity of the TCR, which is generated by recombination of specific variable (V), diversity (D) and joining (J) segments in the case of TCR β and δ, and unique V and J segments for TCR α and γ. Complementarity determining regions (CDR) are the most variable part of the TCR and complement an antigen–MHC's shape. The CDR is divided into three regions termed CDR1–3, and of these CDR1 and CDR2 are coded for by the V segment, whereas CDR3 incorporates a part of the V segment and the D as well as the J segments for TCR β and parts of the V and J segments for TCR α. CDR3 is the most variable region and interacts with the target oligo-peptide lodged in the antigen-binding groove of the HLA molecule of an antigen-presenting cell [2]. The germ line TCR β locus on chromosome 7q34 has two constant, two D, 14 J and 64 V gene segments, which are recombined during T-cell development to yield numerous VDJ recombined T-cell clones; likewise, TCR α locus on chromosome 14q11 has one constant, 61 J and 44 V segments (http://www.imgt.org/IMGTrepertoire/LocusGenes/index.html#C). Further variability and antigen recognition capacity is introduced by nucleotide insertion (NI) in the recombined TCR α and β VDJ sequences. This generates a vast T-cell repertoire, yielding in excess of a trillion potential TCRαβ combinations capable of reacting to non-self (and self) peptides [3]. Since the advent of next generation sequencing techniques, the TCR repertoire, as estimated by TCR β clonal frequency measurement has revealed that the T-cell repertoire in healthy individuals is complex with thousands of clones in each individual spanning a spectrum of high and low frequencies [4,5].

T cells have a fundamental role in clinical medicine, especially in cancer therapeutics. As an example, clinical outcomes following stem cell transplantation (SCT) are closely associated with T-cell reconstitution, both from the standpoint of infection control and control of malignancy [6,7]. T-cell reconstitution over time following SCT may be considered as a dynamical system, where T-cell clonal expansion can be modelled as a function of time using ordinary differential equations, specifically the logistic equation. This suggests that successive states of evolution of T-cell repertoire complexity when plotted as a function of time may be described mathematically as a deterministic process [8,9]. Support for determinism shaping the T-cell repertoire in humans comes from the observation of fractal self-similar organization with respect to TCR gene segment usage [10]. Fractal geometry is observed in structures demonstrating organizational self-similarity across scales of magnitude, in other words structures look similar (not identical) no matter what magnification they are observed at. This structural motif is widely observed in nature, e.g. in the branching patterns of trees and in the vascular and neuronal networks in animals [11–14]. However, while mathematical fractal constructs may be self-similar over an infinite number of scales; in nature, the scales of magnitude demonstrating self-similar organization are limited. Mathematically, logarithmic transformation of simple numeric data is used to identify this scale invariance, because this makes values across different scales comparable. Self-similarity in fractals is evident if the logarithm of magnitude of a parameter (y) maintains a relatively stable ratio to the logarithm of a scaling factor value (x), a ratio termed fractal dimension (FD) [15]. FD takes on non-integer values between the classical Euclidean dimensional values of one, two and three used to define the dimensions of a line, square and a cube. Fractal geometry has been used to describe molecular folding of DNA, and the nucleotide distribution in the genome [16–19]. In such instances, FD explains the complex structural organization of natural objects. Evaluating T-cell clonal frequencies, when unique clonotypes bearing specific TCR β J, V + J and VDJ + NI are plotted in order of frequency, a power law distribution is observed over approximately 3–4 orders of magnitude. This proportionality of clonal frequency distribution across scales of magnitude (number of gene segments used to define clonality in this instance) means that there are a small number of high-frequency clones, and a proportionally larger number of clones in each of the lower frequency ranks in an individual's T-cell repertoire [10,20].

The observed determinism of the TCR repertoire poses the question as to whether this may originate in the organization of the TCR locus, and whether this may also be described mathematically. Using fractal geometry, one may consider the TCR loci similarly, such that when the linear germ-line DNA of the TCR V, D and J segments is rearranged, this process lends geometric complexity to the rearranged locus compared to its native state, in other words, changes its FD. Another feature of the TCR gene segment distribution arguing against the stochastic nature of TCR gene rearrangement is the periodic nature of their location on the gene locus. Repetitive or cyclic phenomenon too may be described mathematically using trigonometric ratios. It is to be noted that the universal constants π and e have a role in calculating the trigonometric ratios and FD, and have been identified in analysis of the T-cell repertoire [10], which would imply that the TCR locus may be arranged in conformity with these constants, and demonstrate self-similarity as well as periodic characteristics. An inspection of the known sequence of the TCR loci with attention to V and J gene segment size and spacing supports this notion. In this theoretical paper, an examination of the TCR α and β loci, in light of e and π is presented in support of the hypothesis that TCR locus organization may influence T-cell repertoire.

2. Material and methods

2.1. Human T-cell receptor loci

Data on the T-cell receptor α/δ (TRA) and β (TRB) loci were obtained from the NCBI, using the public database (http://www.ncbi.nlm.nih.gov/nuccore/114841177?report=graph; http://www.ncbi.nlm.nih.gov/nuccore/99345462?report=graph) (electronic supplementary material, tables S1 and S2). The graphic format, with locus identification was used. Data collected included the position of the initial (start; first nucleotide of exon 1 for V gene segments) and final (stop: last nucleotide of exon 2 for V gene segments) nucleotides of all the TCR gene segments, beginning from the centromeric (5′ end) of the locus and going to the telomeric (3′ end). Segment length was taken from the NCBI database, and the intergenic spacing between consecutive segments was calculated by taking the difference between the numeric value of the initial nucleotide (or base pair) position (x_f) of a gene segment and the initial nucleotide position (x_i) of the following segment (electronic supplementary material, tables S1 and S2). In the ensuing calculations, numeric data were either considered without any modification or transformed to natural logarithms to eliminate the effect of relative magnitude between the variables being examined, and also to allow comparisons across different scales, e.g. segment size in hundreds or tens of nucleotides versus intergenic spacing in the thousands or hundreds of nucleotides, respectively. Microsoft Excel (v. 14.2.5) was used to perform various calculations presented in this paper.

2.2. Gene segment distribution on the T-cell receptor locus

An examination of the TCR gene segments on the TRA and TRB loci reveals that the V segments are longer in length and spread farther apart than the J segments which are shorter and clustered together more closely (http://imgt.org/IMGTrepertoire/index.php?section=LocusGenes&repertoire=locus&species=human&group=TRB; http://imgt.org/IMGTrepertoire/index.php?section=LocusGenes&repertoire=locus&species=human&group=TRA). This suggests that the sets of V and J gene segments in both the TRA and TRB loci are self-similar in their geometry, that is, they have the same distribution albeit at different scales of magnitude. To investigate the possible fractal nature of the TCR locus, as the basis for a deterministic TCR gene rearrangement paradigm, individual V and J gene segments and the adjacent intergenic segment together were considered line segments. Gene segments were chosen as scaling factor, because of their relatively uniform length (in base pairs), and because the gene segments are the coding regions brought together in TCR gene rearrangement. In other words, if the TCR loci are represented by strings of beads, the number of beads will determine the length of the string. The fractal geometry of the locus may then be investigated by examining the spatial distribution of contiguous V and J segments and the adjacent intergenic space separating them. One way to accomplish this will be to examine the relationship between magnitude and scaling factor across varying levels of magnification (sets of V gene segments versus sets of J gene segments) to calculate the FD

To determine the FD of the TCR loci, the FD was calculated analogous to the calculation of the FD of a Koch snowflake curve [15]; the length of the gene segment in number of nucleotides or base pairs is considered the scaling factor and the length of the gene segment + intergenic space, also in nucleotide number, the magnitude of the line segment. This calculation was performed individually for each V and J segment to fully account for variability observed.

2.3. T-cell receptor β locus periodicity

The TCR gene segments occur periodically from the 5′ to the 3′ end of the loci, with V, (D in TRB and TRD), J and C segments, generally in that order. For the calculations regarding the gene segment periodicity and its influence on gene usage frequency, the helical DNA molecules were considered as a propagating spiral (or a wave). In this model, each basepair on a strand of DNA may be considered as a point x, with subsequent base pairs, x + 1 … x + n being successive points on the spiral, as opposed to points on a straight number line. The spiral or helical DNA molecule, as it executes one turn goes through approximately 2π radians, in terms of angular distance spanned. One turn of the helix contains 10.4 nucleotides [21], so the space between successive nucleotides may be considered as the angular distance in radians between them (assuming a uniform unit radius of the DNA molecule). This inter-nucleotide ‘distance’ will be 2π/10.4 (electronic supplementary material, figure S1). The spatial position of any nucleotide x, relative to the locus origin may be then be described as the angular distance in radians ((2πx/10.4) radians) and its coordinates on the DNA molecule determined. This measure may then be used to determine the relative position of the various TCR gene segments.

2.4. T-cell clonal frequency

SCT donor and recipient samples for determining T-cell clonal frequency were obtained as part of a clinical trial approved by the institutional review board at Virginia Commonwealth University (ClinicalTrials.gov Identifier: NCT00709592). As previously described, CD3+ cells were isolated from SCT donor samples and cDNA synthesized from these cells [10]. The cDNA was then sent to Adaptive Biotechnologies (Seattle, WA) for high-throughput sequencing of the TCR β CDR3 region using the ImmunoSEQ assay. This approach comprises a multiplex PCR and sequencing assay in combination with algorithmic methods to produce approximately 1 000 000 TCR β CDR3 sequences per sample [22]. The assay used 52 forward primers for the Vβ gene segment and 13 reverse primers for the Jβ segment, to generate a 60-bp fragment capable of identifying the entire unique VDJ combination [23]. Amplicons were then sequenced using the Illumina HiSeq platform, and data were analysed using the ImmunoSEQ Analyzer set of tools. This approach enables direct sequencing of a sample of the circulating T-cell populations, representing a fraction of the TCR repertoire. The data output for this assay are the frequency of unique CDR3 sequences, defining the TRB V, D and J segments used and the N substitutions in each rearranged sequence [10]. This permits estimation of the relative frequency of each gene segment in the sample T-cell population. To estimate specific gene-segment usage, the copy number of CDR3 sequences containing unique V or J gene segments were summed and used in the subsequent calculations. Therefore, to determine the relative contribution of the ith TRB V segment (fVi) to the repertoire, the proportion of the sequence copy number for that segment to the sum of all the sequence reads for all the TRB V segments was determined. This clonal frequency estimate for each V segment was then compared with a calculated expected proportion if all the V segments contributed equally to the repertoire. This was determined by averaging the sum of all frequencies over all the possible TRB V gene segments to give a simulated clonal frequency— for each TRB V segment, which calculates to 1.49%. Student's t-test was used to determine the statistical significance of expected versus measured clonal frequencies for each TRB V segment.

3. Results

3.1. Self-similarity of the T-cell receptor loci

Self-similarity across the TRA and TRB loci was first examined by deriving the FD of these loci. Given the relative uniformity of sizes (in nucleotides) for the V and J gene segments, these were used as scaling parameters for determining the FD of the TCR loci (FD-TCR). The gene segments and the intergenic spaces together were considered the magnitude of the TCR locus line segment for this calculation. The sizes of each segment and intergenic space were used in the FD-TCR calculations to account for variability across the locus. The following formula was used:

3.1

Relatively consistent values of FD-TCR were observed across the distribution of the V and J segments for both the TCR α and β loci, when the calculated FDs were plotted across the loci. Therefore, the regions of the locus bearing the V genes may be considered as a magnified version of the J segment-bearing regions for both TRA and TRB (figure 1). This indicates that despite the differences in scale of the TCR regions bearing these gene segments, when viewed on a logarithmic scale, there is uniformity in the size and distribution of gene segments both within and between the different TCR loci; a hallmark of self-similar systems. The average FD-TCR of the TRB V and J segments were 1.4 ± 0.1 and 1.3 ± 0.2, respectively. Corresponding values for the TRA locus were 1.5 ± 0.1 and 1.7 ± 0.1, for the V and J segments. The self-similarity across the TRA locus may also be seen when the spacing between successive gene segments is plotted from the 5′ to 3′ end in a circular area graph (figure 2). The two halves of the figure are similar in appearance and are symmetric, except for the spacing between TRA-J segments being an order of magnitude lower when compared with TRA-V segments. This implies that there exists spatial symmetry in the TCR loci bearing different T-cell gene segments (V and J specifically), evident in the proportionality of the size and distribution of the gene segments. It may be hypothesized that this phenomenon exerts an influence on the order of TCR gene rearrangement, such as the Dβ to Jβ and DJβ or Jα to Vβ or Vα recombination.

Figure 1.

(a) TCR α (TRA) and (b) TCR β (TRB) FD calculated for each variable and joining segment (equation (3.1)). Values at each point are plotted along the nucleotide positions on the locus. Inset in each figure shows the values for the J segments in comparison with the entire locus. Note scale difference of approximately 1 log. TCR δ region on the TRA locus is excluded, as are the D and C segments for the TRB locus.

Figure 2.

Relative position of the first nucleotide of each TRA gene segment from the 3′ end (blue) of the locus plotted against spacing (red) following that TCR gene segments (y-axis, log₁₀-scale). This demonstrates self-similarity in the gene segment size and spacing distribution across the V (right) and J (left) loci, with the two halves of the figure demonstrating symmetry. Log scale used.

In calculating FD-TCR, the D and C segments for TRA and TRB and the V30 segment of TRB were not considered because of their infrequent and non-periodic occurrence, as well as being interspersed between other loci. However, notably their size followed fixed proportion to the sizes of the J and V segments, respectively, such that D segments were approximately 1/3 to 1/4 the size of J, and the C segments were approximately three times the length of the V segments. Further, the TRB V30 segment did follow similar rules in terms of magnitude and intergenic space length (from the adjacent C segment located 5′ of V30) as previously recorded for other V segments. Similarly, the TCR-γ locus was also not evaluated because of the small number of gene segments, however, it is to be noted that the gene segment and intergenic lengths are similar to the TRA and TRB loci.

During TCR recombination, the J segments in the TRA locus and the D segments in the TRB locus, are recombined with the V segments. The spatial distribution of V segments in the TRA and TRB loci was therefore examined relative to the position of the J and the D loci, respectively, to determine the spatial relationship between the recombining loci. The relative position of V loci was calculated with respect to the two D segments for TRB, and the 5′-, mid-point and 3′-J segments for TRA, by a formula considering the 5′-initial nucleotide positions (x_i) of the D or J gene segments, and the final, 3′ nucleotide position of the V segments (x_f) from the origin of the locus

3.2

Relative recombination distance (RRD) demonstrated a logarithmic decline as more telomeric V segments were considered across both TRA and TRB loci. When RRD for successive TRB V loci was plotted in order of occurrence on the locus, the value declined as a logarithmic function of V segment position, with a slope of 1.6 for the TRB locus (electronic supplementary material, figure S2a) and approximately 1.3 for the TRA locus (electronic supplementary material, figure S2b,c). The consistency observed between the slope of decline in the V segment distance from the D or J segments and the previously calculated FD-TCR supports the notion of self-similarity of the TCR loci as seen in the preceding calculations.

3.2. Logarithmic scaling of the T-cell receptor gene segment periodicity

In the self-similarity analysis, the FD-TCR oscillates around a central value with regular periodicity. Further, the repetitive occurrence of gene segments on the TCR loci, suggests that they conform to a periodic distribution analogous to the cyclic behaviour exhibited by phenomenon such as wave motion, or in this case DNA helix/spiral progression. To examine the periodicity of the relative positions of gene segments on the TCR loci, they were considered as successive nucleotide sequences on the DNA helix and the angular distance between segments determined by using the relationship 2πx_i/10.4, where x_i is the initial or final nucleotide position of the ith gene segment with respect to the TCR locus origin (electronic supplementary material, figure S1). The calculated angular distance between the gene segments was further analysed by determining the distance between V and D segments in TRB. This was measured from the 5′, centromeric-end of the D segments to the 3′, telomeric-end of the V segments. These values were used to determine the coordinates of the gene segments on the DNA helix, using the trigonometric parameters, sine and cosine for the initial nucleotides (x_i) relative to the locus origin. This was done for the angular distance, AD = 2πx_i/10.4, and the resulting sine and cosine values for the nucleotide positions plotted against the angular distance (f(AD) = sin (2πx_i/10.4) or cos (2πx_i/10.4)) from locus origin. No clear pattern was discernable, with the sine and cosine values for each of the positions distributed randomly along the length of the TCR DNA strand (figure 3a). Given the previously observed logarithmic ordering of the TCR loci, the trigonometric functions of the natural logarithm of 5′-initial nucleotides angular distance from the locus origin were then plotted against the angular distance, f(AD) = (sin (log (2πx_i/10.4)) and (cos (log (2πx_i/10.4)). This demonstrates that the TCR gene segments were positioned on the locus in a sequentially aligned manner (figure 3b) for both the TRA and TRB loci when the log-periodic functions of the nucleotide positions are considered. Further, when the TRA and TRB sine and cosine plots were overlaid, these plots were superimposable, supporting the earlier observation regarding the self-similar nature of these TCR loci (figure 3c). The TRD gene locus is similarly organized, even though it is nested in the TRA locus (figure 3d). These calculated trigonometric functions of TCR gene segments reveal that these loci have a log-periodic nature.

Figure 3.

Logarithmic ordering of periodic TRB gene segments. (a) Angular coordinates, i.e. sine (blue diamonds) and cosine functions (red) of TRB gene segment 5′ initial nucleotide's angular distance from locus origin (5′ end) plotted across the TCR loci. The x-axis depicts angular distance of gene segments from origin. (b) Sine (orange for TRB; blue for TRA) and cosine functions (green for TRB; red for TRA) of the natural logarithm of TRB gene segment 5′ first nucleotide and TRA 3′ last nucleotide angular distance from locus origin (5′ end) plotted across the TCR loci. Angular coordinates for TRD gene segments (cosine, orange circle; sine, blue circles) within the TRA locus depicted in the third graph. (c) TRA and TRB sine and cosine coordinates superimposed, demonstrating self-similarity across the two loci.

3.3. T-cell clonal frequency and its correlation with T-cell receptor β gene segments

To determine whether the periodicity of the TCR loci was reflected in the T-cell gene segment usage and subsequent clonal repertoire observed in humans, high-throughput TRB sequencing data from six HLA matched unrelated SCT donors and recipients were used. The samples were obtained from the donor apheresis product and from patient blood, either at or beyond day 100 following SCT. Clonal frequency estimates, expressed in copy numbers of unique sequence reads for TRB V and TRB J segment containing clones were plotted against the calculated angular distance (in radians) between the 5′ initial nucleotide (2πx_i/10.4) of the TRB-D1 (and TRB-D2) and the 3′ final nucleotide (2πx_f/10.4) of the V segments or the 5′ start nucleotide of the J segments. The TRB gene segment usage in the T-cell repertoire varied as a quasi-periodic function of the angular distance between both TRB-D1 and D2 and the successive TRB-V segments, oscillating between high and low clonal frequency values (figure 4a). Further, the two J segment-bearing regions of the TRB locus were approximately 5000 radians apart (in the 3′ direction) and also demonstrated oscillating clonal frequency (figure 4b). This finding was consistent between all six unrelated stem cell donors and transplant recipients following SCT, demonstrating very high expression levels for some loci, intermediate for others and low or no expression in others (figure 4c). To determine the relative likelihood of various V segments being involved in TCR rearrangements, the total clonal frequency (total copy number of all the TRB sequences) in each donor was averaged across all the V segments, and the measured clonal frequency for each V segment was compared with this average (table 1). This analysis demonstrated a consistent, significantly variable rate of recombination for several TRB V gene segments (both higher and lower than the predicted average) supporting a role for the periodic organization in determining the TCR repertoire genesis. This periodicity may be interpreted as TCR gene V-segment recombination probability amplitude oscillating between 0 (no recombination) and 1 (very frequent recombination) across the locus, resulting in either low or high gene segment usage in the resulting T-cell clones. It should be noted that the clonal frequency estimates reported here must be interpreted with caution because our data are based on high-throughput sequencing of T-cell cDNA rather than genomic DNA, which may give a closer estimate of clonal frequency [24]. Further, the calculations used do not report the number of unique CDR3 sequences with specific TRB gene segments, instead give the sum of all the CDR3 sequences with the specific V and J gene segments in blood samples from the donors and recipients. As such this method does not take into account T-cell clonal expansion, which partially contributes to the higher copy number of individual TCR gene segments. However, a logical interpretation of these data is that, if a particular segment is encountered at a high frequency in an individual, it will also be more likely to have been involved in the recombination and thus have a larger number of clones represented in the T-cell repertoire. Further, the consistency observed in the frequency of specific TRB V and J segment usage across several unrelated donors and patients reconstituting T cells following SCT provides greater confidence in the reproducibility of this observation.

Figure 4.

Estimated TCR gene segment usage frequency as a periodic function of the V and J gene segment positions on the TRB locus. (a) TRB-V containing sequence frequency demonstrates quasi-periodicity (irregular) as a function of angular distance between TRB-D1 and successive TRB-V segments from V29 to V1, and may represent periodic fluctuation in functional recombination probability amplitude across the locus. (b) TRB-J containing sequence frequency as a function of the angular distance between TRB-D1 and successive TRB-J segments. Both x- and y-axes magnitude differ by approximately an order of magnitude (one-log) between the two graphs. (c) Similar periodicity is observed in V segment containing sequence frequency when unique pre-transplant donor and post-transplant recipient samples are analysed. Gene segment frequencies; donor—blue; post-SCT patient—red.

Table 1.

Per cent contribution of each TRB V gene segment to the T-cell repertoire in six normal volunteer unrelated stem cell donors. Data derived from copy number of specific TRB V segment containing sequences identified by high-throughput TRB sequencing of cDNA from CD3+ cells from GCSF mobilized unrelated stem cell donor blood. Significance values were calculated by comparing each data point with the expected contribution of each V segment if it were to contribute equally to the repertoire; calculated at 1.492% for each V segment. Asterisks denote significant positive or negative variation from expected average % contribution.

TRB-V	TRB-D1 to Vn	P3	P5	P7	P8	P9	P14	t-test p
V1*	331 241	0	0	0	0	0	0
V2	330 045	4.0303	2.7644	2.4012	3.0766	4.0050	3.0912	0.0014*
V3-1	325 440	1.0648	1.0226	0.2479	0.7234	0.6134	0.5356	0.0015*
V4-1	322 650	0.9695	2.8362	2.7299	2.2634	1.0630	1.7430	0.2393
V5-1	317 898	6.3966	6.9114	12.1711	8.4674	7.5652	8.1077	0.0005*
V6-1	313 522	2.5084	0.9548	1.1643	1.1733	2.6344	0.6788	0.9412
V7-1*	311 156	0.0247	0.0124	0.0255	0.0101	0.0905	0.0124	0.0000*
V4-2	303 133	0.6544	6.1978	1.0494	5.4131	1.0694	3.1484	0.2061
V6-2	300 838	0	0	0	0	0	0
V3-2*	294 791	0	0	0	0	0	0
V4-3	292 705	1.0170	6.8112	3.4859	5.1722	0.9413	4.4701	0.0732
V6-3	287 739	0	0	0	0	0	0
V7-2	285 506	4.5615	3.2010	5.7065	2.3890	6.8307	3.7282	0.0075*
V8-1*	281 943	0	0	0	0	0	0
V5-2*	273 484	0	0	0	0	0	0
V6-4	268 439	2.2044	0.6536	1.3542	0.9932	0.6786	1.3933	0.2907
V7-3	266 269	2.5632	1.1765	4.1219	2.9028	3.8841	3.9316	0.0177*
V8-2*	265 178	0	0	0	0	0	0
V5-3*	263 352	0.0026	0.0058	0.0056	0.0071	0.0035	0.0065	0.0000*
V9	261 694	3.2478	8.0157	3.8229	8.1913	2.9329	4.5373	0.0133*
V10-1	256 881	1.8050	0.7055	0.5047	0.0141	0.5434	0.1277	0.0201*
V11-1	252 177	0.1184	0.1115	0.1345	0.1757	0.0887	0.1097	0.0000*
V12-1*	247 647	0	0	0	0	0	0
V10-2	241 738	0.2480	0.2294	0.1190	0.0379	0.1669	0.0962	0.0000*
V11-2	236 331	4.5326	1.9117	1.8709	1.7148	1.1813	1.6928	0.2354
V12-2*	232 144	0	0	0	0	0	0
V6-5	226 028	0	0	0	0	0	0
V7-4	223 500	0.0667	0.0188	0.0311	0.0426	0.0807	0.0464	0.0000*
V5-4	218 697	2.2039	3.2502	2.3698	3.2183	2.8193	2.9704	0.0007*
V6-6	214 807	0	0	0	0	0	0
V7-5*	212 056	0	0	0	0	0	0
V5-5	206 949	0.9916	1.0526	1.1532	1.3039	0.9801	1.8838	0.1173
V6-7*	203 762	0.0217	0.0034	0.0079	0.0160	0.0221	0.0097	0.0000*
V7-6	201 119	1.0252	0.4609	0.8142	0.4572	0.9186	0.5826	0.0005*
V5-6	196 369	1.8120	2.8221	1.8170	2.6974	1.4738	2.8758	0.0298*
V6-8	191 973	0.0165	0.0113	0.0117	0.0149	0.0176	0.0098	0.0000*
V7-7	189 367	0.3461	0.1962	0.2955	0.2034	0.3040	0.1802	0.0000*
V5-7	184 275	0.0011	0.0007	0.0012	0.0018	0.0022	0.0013	0.0000*
V6-9	179 882	0.0251	0.0026	0.0077	0.0104	0.0160	0.0020	0.0000*
V7-8	177 063	0.9882	1.2872	1.0373	0.9781	1.5338	1.0746	0.0123*
V5-8	172 110	0.1002	0.1722	0.1302	0.1331	0.0890	0.1466	0.0000*
V7-9	166 547	6.6246	3.1945	5.0938	3.8699	5.0317	2.7584	0.0040*
V13	162 602	1.3318	2.4366	1.1211	1.6865	1.5521	0.8024	0.9863
V10-3	157 540	6.8222	3.0340	2.4871	3.0834	4.8708	2.1408	0.0271*
V11-3	151 115	0.1682	0.1030	0.2749	0.4546	0.2184	0.5007	0.0000*
V12-3	147 725	0	0	0	0	0	0
V12-4	145 717	0	0	0	0	0	0
V12-5	135 331	0.8487	0.1232	0.1602	0.5811	0.4903	0.5099	0.0002*
V14	131 145	2.7199	0.8741	0.3784	1.2145	2.7262	0.7814	0.9211
V15	128 057	1.6939	3.6575	1.5654	3.8924	1.2678	2.7474	0.0873
V16*	125 040	0.0530	0.3326	0.2467	0.3983	0.0670	0.1695	0.0000*
V17	122 686	0.0000	0.0003	0.0000	0.0001	0.0000	0.0001	0.0000*
V18	114 255	3.4473	2.1660	3.4115	2.5263	3.2758	2.3010	0.0024*
V19	112 371	7.2938	2.6726	5.0988	3.0702	10.1580	3.3294	0.0256*
V20-1	107 619	7.1806	10.7685	13.3216	10.7729	7.9844	14.3099	0.0005*
V21-1*	101 590	0	0	0	0	0	0
V22-1*	98 688	0	0	0	0	0	0
V23-1*	96 107	0.3758	0.4921	0.8236	0.6453	0.8016	0.8811	0.0002*
V24-1	90 480	0.9017	0.4644	0.2781	0.3573	0.9437	0.3228	0.0006*
V25-1	81 801	0.0311	0.0090	0.0025	0.0099	0.0248	0.0077	0.0000*
VA*	75 388	0	0	0	0	0	0
V26*	66 734	0	0	0	0	0	0
VB*	57 218	0	0	0	0	0	0
V27	54 838	0.8159	0.4393	0.1291	0.4106	0.7962	0.3384	0.0003*
V28	51 638	3.3656	4.0760	1.4496	3.5903	6.8305	7.3979	0.0236*
V29-1	39 705	10.9788	9.2792	10.9956	9.2776	9.0522	10.3019	0.0000*
V30	−12 569	1.7997	3.0778	4.5697	2.3565	1.3586	3.2055	0.0465*

4. Discussion

The germ-line genomic TCR (and immunoglobulin) loci have the unique characteristic in that these can undergo DNA double strand break and recombination resulting in VDJ rearrangement which results in the generation of numerous unique T-cell (and B-cell) clones with the ability to identify a wide array of antigens. In this analytical manuscript, characteristics of the organization of TCR gene segments are explored, and spatial symmetry identified across regions of these loci. Further, the possible relationships between the periodic occurrence of the TCR gene segments and its potential influence on T-cell clonal frequency are also investigated. It is found that the TCR loci are organized with mathematical precision, which suggests that T-cell repertoire formation may be a deterministic process.

TCR rearrangement is an ordered process, with the sequence of recombinatorial events governed by DNA sequence motifs occurring at regular frequency in the TCR loci. The so-called ‘12/23 rule’ is an example of this, where the TRB V, D and J segments are flanked by conserved sequences called recombination signal sequences (RSS), comprised a heptameric and a non-americ sequence, which are interposed by either a 12 (RSS-12) or a 23 (RSS-23) base pair sequence. VDJ rearrangement is brought about by recombinase-activating gene-1 (RAG-1) and RAG-2 protein complexes, which always bring together segments with a RSS-12 with a segment flanked by a RSS-23, not otherwise. Considering the TRB, Dβ segments are flanked by the RSS sequences on both sides (5′, RSS-12 and 3′, RSS-23), Jβ segments on the 5′ end (RSS-12) and Vβ segments on the 3′ end (RSS-23) [25]. In the series of events set off during T-cell development, initially a Dβ segment rearranges with one of the Jβ segments, then the combined DJβ joins with a specific Vβ segment to result in VDJβ rearrangement, yielding a unique T-cell clone. Similar considerations hold for TCRα, where Jα recombines with Vα, with the additional possibility of locus editing such that alternative 5′ Vα may be rearranged to an alternative 3′ Jα segment at a later time [26]. As elegant as this system is, it does not completely explain either how the order of recombination is determined or what determines the variability in the use of various TCR gene segments encountered in the T-cell repertoire; for instance, why does the Dβ RSS-23 preferentially rearrange with Jβ RSS-12, and not the Dβ RSS-12 with the Vβ RSS-23, and, why are clones with TRB V5-1 encountered much more frequently than TRB V5-4 in the normal T-cell repertoire? Further, under normal circumstances, in the rearrangement of the TRD (TCR-δ) locus, some V segments in this and the TRA locus recombine only with TRD-D, while others recombine both with TRD-D and TRA-J segments [27], despite the TRD locus being nested within the TRA locus. Further, when a TRA-Jα segment is ectopically introduced in the TRD locus at the TRD-Dδ segment position, rearrangement of this ectopic TRA-J with TRD-V segments occurs, demonstrating that locus position, rather than the actual sequence may have a deterministic role in the order of recombination [28]. This raises the question of whether there are further structural features of the DNA molecule, which determine the order of recombination. It is this question that the log-periodic nature of the TCR locus elucidated in our analysis may help answer.

In nature there is a tendency for organizational patterns to be repeated over different scales of measurement and for such patterns to be observed across different systems. Fractal organization in the VDJ segment usage in the T-cell repertoire of normal individuals has been observed with the diversity, joining and variable gene segment usage defining a virtual ‘structure’ that results from recombination of the T-cell β receptor locus [10,20,29]. With this background, the proportions between the V and J segment size and intergenic segment lengths between adjacent segments were examined relative to each other and found to be similar, demonstrating spatial symmetry between the TCR regions harbouring the V and J segments. It is likely that the proportional distribution of V and J segment size and spacing between individual segments (fractal organization) in this instance serves to order the ensuing rearrangement process. This may in part explain why in the order of gene segment rearrangement, Dβ to Jβ and DJβ or Jα to Vβ/α segments, RAG complexes are always directed from the shorter, closely spaced J segments to the longer, more dispersed V segments, such that the reverse does not transpire in the course of normal recombination. Further, the logarithmic scaling implies that the distribution of these size-ordered segments is always similar in their respective sections of the TCR locus, which ensures that RAG complexes do not have to ‘scan’ an entire sequence of nucleotides to randomly encounter a coding segment, but can potentially align with relevant segments, skipping over given lengths of intergenic material. This would then provide an additional mechanism to complement the 12/23 rule and ensure fidelity of recombination. Epigenetic mechanisms such as RAG2 interacting with methylated histone H3-K4, further facilitates the VDJ recombination [30]. Other sequence motifs critical in terms of facilitating VDJ recombination are the ubiquitous CTCF-cohesin-binding GC-rich consensus sequences [31,32]. These trans-acting factors help bring about conformational changes in the locus, which bring V segments in apposition to J segments allowing successful recombination. However, while they give important mechanistic insights, the sequence motifs and chromatin-based landmarks for recombination still require appropriate scaling—logarithmic—as in the measurements presented here, to yield a quantifiable effect on the TCR recombination process. This hypothesis, if true, suggests that the origin of the fractal properties of the T-cell repertoire clonal distribution is within the arrangement of the TCR loci resulting in an ordered recombination process. The log-periodic nature of other fractal phenomenon encountered in nature supports this postulate [33,34].

High-throughput sequencing of TRB has demonstrated a differential representation of the different gene segments in the T-cell clonal repertoire, indicating that some sequences are used at a higher frequency than others [4,5,10]. This has been observed for TCRγ as well as TCRβ and has been seen for both J and V segments [35]. This recombination bias affects both in-frame and out-of-frame recombined sequences, suggesting that it is not a consequence of thymic selection, nor HLA restriction, but rather is a result of recombinatorial usage bias, or ranking of various segments. Figure 4 demonstrates this phenomenon, and it is also reflected in the power law distribution of the final T-cell clonal distribution observed. The relationship between TCR locus organization and segment selection in this rearrangement process and its impact on the T-cell repertoire generation has been a focus of intensive study in the recent years. Recently, a biophysical model describing yeast chromosome conformation has been applied to the murine TCR β-D and -J segment and the derived model based on ‘genomic distance’ between these segments has partially recapitulated the observed bias in J segment usage [36]. This supports the notion that chromatin conformation, and TCR spatial organization has a formative role in the T-cell repertoire generation. Regardless of the mechanism of recombination, it has become obvious that the T-cell repertoire that emerges has a ‘biased’ VDJ segment usage, with certain segments being used more frequently than others. This suggests that these segments may be more efficiently rearranged resulting in their over representation in the repertoire and vice versa. The effect of spatial organization of TCR gene segments on recombination frequency is also evident when modelling the rearrangement likelihood in the murine TRA taking into account the relative positioning of V and J segments [37]. Assuming sequential availability of V and J segments to recombine with each other in a time-dependent process, it was demonstrated that the proximal, central and distal J segments had a greater likelihood of recombining with the correspondingly positioned V segments. The model output demonstrates a ‘wavefront’ of recombination probability propagating through each of the regions when individual J segments were analysed for their ability to recombine with the V segments and vice versa. A similar model examined the recombination probabilities as a function of the size of the ‘window’ of the TRA-V and -J regions available, putting forth the notion that sequential availability of individual gene segments determines the recombination frequencies [38]. These models reinforce the deterministic aspect of the TCR locus recombination and highlight the importance of the scaling observations we report in this paper.

Given the emergence of the constant π in the equations describing the fractal nature of the T-cell repertoire in normal stem cell donors and the periodic nature of TCR gene segments on the TCR locus, their relative positions were examined using trigonometric functions to account for the helical nature of DNA. Similarity was observed in the relative location of the V, D and J segments across the TRA and TRB loci when they were examined using logarithmic scaling, with increasingly complex waveforms observed as higher-order harmonics were evaluated (data not shown). There are several important implications of this observation. First, analogous to the phenomenon of superposition (constructive or destructive interference) observed in the mechanical and electromagnetic waves, one may consider that relative position of a particular segment, reflected by the coordinates on the DNA helix (estimated by the sine and cosine functions, and angular distance from the rearranging segment, Dβ or Jα), may influence its usage in repertoire generation resulting in the periodic distribution of the V and J segment usage in T-cell clones when the locus is interrogated from the 5′ to 3′ end. Essentially, this means that using analytical techniques such as Fourier's series, probability amplitudes may be determined for the various gene segments on the TCR loci based on their positions. It may be very likely that the recombination is most frequent for gene segments that occur at a certain ‘harmonic’ frequency. As an example in the data presented, the TRB-V segment clonal frequency appears to oscillate with a wavelength of approximately 50–60 000 radians from the TRB-D segment (figure 4). This organizational pattern is also observed in the distribution of V gene segments capable of recombining with TRD-D segments, which are approximately 100 000 radians apart on the TRA locus, scattered among the TRA-specific V genes (figure 3). It may be speculated that the gene segment distribution periods represent optimal energy distribution for recombination to occur on the long helical DNA molecule, analogous to the interference phenomenon encountered in wave mechanics. This is plausible because the DNA double helices may represent two superposed waves, and the gene segment location may lend itself to either constructive or destructive interference, impacting the interaction with RAG enzymes and recombination potential. This would in turn determine the probability amplitude of that gene segment being represented in the final T-cell clonal repertoire (figure 5). Evidence to support a role for varying energy distribution along the DNA molecules is beginning to emerge as, such as, in modelling electron clouds of DNA molecules as chains of coupled harmonic oscillators have demonstrated the association between the quantum entanglement in the electron clouds of DNA molecules and the local binding energy [39]. It has also been demonstrated that the lower energy requirements for bending and rotation of the CG-rich DNA sequences, allows more efficient bending of DNA molecules around histones, resulting in greater CG content around nucleosomal DNA [40].

Figure 5.

A model depicting TCR organization and its influence on gene segment recombination probability. TCR V segments are separated by long intervals, J segments by shorter intervals (dashed lines); the ratio of log segment length to log spacing is approximately 1.4 for V segments and approximately 1.3 for J segments. Relative interval between successive V segments and the J segments in the TRA locus (top blue curve) declines logarithmically with a slope of approximately 1.3. Sine and cosine function value of the start nucleotides of each V segment extrapolated to the sense (green) and antisense (blue) DNA strands, demonstrate that the gene segments are accurately aligned once the logarithmic organization is accounted for. Hypothetically, the segment location on the two DNA helices being in-phase or out-of-phase may impact the energetics of DNA–RAG enzyme interaction and thus the probability amplitude (orange line, going from 0 to 1) for gene segment recombination analogous to wave interference phenomenon. In the model depicted, V1 location on the two helices is out of phase, V2 is partially in phase and V3 is completely in phase. Closely clustered together J segments are more likely to be in phase.

In this theoretical paper, we demonstrate that the TCR loci have an iterative, logarithmically scaled periodic nature when the gene segment distribution is considered. The data presented here make it possible to consider variations in DNA recombination of the TCR loci as a partial function of the wave-mechanical properties of the DNA double helix. These findings strengthen the argument that immune responses, such as following SCT may represent an example of an ordered dynamical system.

Authors' contributions

Ab.A.T. collected the data and did most of the calculations reported in the paper. Am.A.T. developed the idea and wrote the paper, as well as performing some of the calculations. M.R. critically reviewed and edited the manuscript. M.H.M. planned and supervised the TRB sequencing and critically reviewed and edited the manuscript.

Competing interests

The authors have no conflicts of interest to disclose.

Funding

Funding for the T-cell sequencing was provided by Genzyme, the manufacturers of Thymoglobulin.