Article Figures & Data
Figures
- Figure 1.
Topographic mapping of the retinocollicular projection in wild-type mice and mice with increased EphA3. S = 2500; a = 1/80; and RE = 36. C, R, Caudal and rostral superior colliculus (SC), respectively; N, T, nasal and temporal retina, respectively. A, Eph receptor density of the normal retinal ganglion cells (EphA3-; thick line R(xR) = S1/2 exp[a (xR - 50)]) and the retinal ganglion cells with increased EphA3 receptor (EphA3+; broken line; R(xR) = S1/2 exp[a (xR - 50)] + RE) on the retina. B, Ligand ephrin density in the SC. L(xL) = S1/2 exp[a (xL - 50)]. C, Topographic maps of the elevated retinal ganglion cells EphA3+ (a) and normal retinal ganglion cells EphA3- (b). xR (percent nasotemporal axis of retina) is the starting point of axonal projection, and xL (percent rostrocaudal axis of the SC) is the arrival point. Topographic map of the retinal ganglion cells (b) is on the line xR + xL = 100. Solid circles, Actual data of the retinocollicular projection of wild-type mice by Brown et al. (2000); open circles, open triangles, actual data of the retinocollicular projection of knock-in mice with EphA3+ and EphA3- retinal axons (Brown et al., 2000), respectively.
- Figure 3.
Influence of effective area of the rostral SC. Critical population density nC = 24, and RE = 36. C, R, Caudal and rostral SC, respectively; N, T, nasal and temporal retina, respectively. A, Topographic map. a, a′, Projection maps of EphA3+ axons; b, b′, projection maps of EphA3- axons. Projection maps on the truncated SC at the rostral side (see D) are indicated a′, b′; a and b are for comparison and the same as a′ and b′ in Figure 2 A. Open circles, open triangles, Experimental results of EphA3+ and EphA3- retinal axons by Brown et al. (2000), respectively. B, C, Population distributions of retinal axon terminals on the rectangular SC and the rostrally truncated SC, respectively. Thick, gray, thin lines, EphA3+ axons, EphA3- retinal axons, and their sum, respectively. D, Effective area of the SC. Shapes of the rectangular and truncated SCs (thick line) are shown. The rostrally truncated SC has a smaller area by one-eighth in comparison with the rectangular SC. Arrows indicate corner positions (horizontal arrow, 0.293; vertical arrow, 35.4).
- Figure 2.
Time course of competition of axon terminals on the SC. Critical population density nC = 24, and RE = 36. A, Topographic map. a to a′, Projection maps of EphA3+ axons; b to b′, projection maps of EphA3- axons. Thin line, dotted line, triangles, thick line, Initial mapping, 5 × 104 steps of competition, 105 steps, and 5 × 105 steps, respectively; open circles, open triangles, experimental results of EphA3+ and EphA3- retinal axons by Brown et al. (2000), respectively. B-E, Population distributions of EphA3+ and EphA3- retinal axons on the SC are on the initial map (B), 5 × 104 steps of competition (C), 105 steps (D), and 5 × 105 steps (E). Thick, gray, thin lines, EphA3+ axons, EphA3- retinal axons, and their sum, respectively. Value of the peak truncated in B is 398.
- Figure 4.
Variation of topographic maps depending on the critical population density nC. Increased EphA3 receptor density, RE = 65. A, Topographic map. Projection maps of nC = 23 (dot), 28 (thick solid line), and 33 (large dot) are presented. Open circles, open triangles, Experimental results of EphA3+ and EphA3- retinal axons by Brown et al. (2000), respectively. B, Population distributions of retinal axon terminals on the SC. Total populations of EphA3+ and EphA3- retinal axons when nC = 23 (dot), 28 (thick solid), and 33 (large dot) are presented.
- Figure 5.
Variation of topographic maps depending on increased EphA3 receptor density RE. Critical population density nC = 28. A, Topographic map. Projection maps of RE = 65 (dot), 50 (large dot), 36 (thick line), and 18 (solid triangle) are presented. Open circles, open triangles, Experimental results of EphA3+ and EphA3- retinal axons by Brown et al. (2000), respectively. Population distributions of retinal axon terminals on the SC when RE = 65 (B), 50 (C), and 18 (D) are presented. For RE = 36, see Figure 3C. Thick, gray, thin lines, EphA3+ axons, EphA3- axons, and their sum, respectively.
- Figure 6.
Computer simulations for topographic maps of homozygous and heterozygous knock-in mice with EphA3 receptor. A, Topographic map of homozygote mice. Simulated projection maps of EphA3+ (a) and EphA3- (b) axons of homozygotes (RE = 36) are presented. Open circles, open triangles, Experimental results of EphA3+ and EphA3- retinal axons of homozygotes by Brown et al. (2000), respectively. B, Topographic map of heterozygote mice. Simulated projection maps of EphA3+ (a) and EphA3- (b) axons of heterozygotes (RE = 18) are presented. Open circles, open triangles, Experimental results of EphA3+ and EphA3- retinal axons of heterozygotes by Brown et al. (2000), respectively. For simulated population distributions of homozygotes and heterozygotes, see Figures 3C and 5D, respectively.
- Figure 7.
Simulations for knock-out mice with disrupted ephrins. A-D, Schematic illustration of the results of experiments of retinocollicular projections in ephrin-A2 and -A5 knock-out mice (adapted from Feldheim et al., 2000). Oval figures show the SC (left, rostral region; right, caudal region). Terminal positions of temporal and nasal retinal axons are shown by solid circles (T) and open circles (N), respectively. A, Wild-type mouse. B, Homozygotes of A2-/-). C, Homozygotes of A5-/-. D, Heterozygotes of A2+/- and A5+/-. E-H, Graded ligand densities in the SC used for computer simulations. Dotted-dashed line (for comparison), Total ligand density in E. E, Solid line, Total ligand density of lines a and b. Line a (dotted line), Ligand density of A2+/+, 27.17(exp[(xL - 50)/82] - 0.35 exp[(xL - 50)/50]). Arrow, Position of broad peak. Line b (broken line), Ligand density of A5+/+, 27.17 × 0.35 exp[(xL - 50)/50]. F, Solid line (line b), Ligand density of A2-/-. G, Solid line (line a), ligand density of A5-/-. Arrow, Position of broad peak. H, Solid line, Total ligand density of A2+/- and A5+/-, 27.17 exp[(xL - 50)/82]/2. I-L, Topographic maps by computer simulations. For the receptor density, the curve similar to that in E (solid line) was used. Thin, gray, thick lines, Initial mapping, 1 × 106 steps of competition, and 3 × 107 steps, respectively; open, solid circles, projected positions of nasal and temporal retinal axon terminals on 3 × 107 steps; arrow, projected position of temporal retinal axon terminal of the initial mapping. I, Map by the ligand density of E (solid line). The map did not change during competition steps. J, Map by the ligand density of F (solid line). Open circles, xL = 100; arrow, xL = 52; solid circle, xL = 21. K, Map by the ligand density of G (solid line). Moderate solid line, Map on 6 × 104 steps; open circles, xL = 71; arrow, xL = 71; solid circle, xL = 21. L, Map by the ligand density of H (solid line). Open circles, xL = 100; arrow, xL = 57; solid circles, xL = 21. M-P, Distribution of population densities of axon terminals on the SC. Thin, gray, thick solid lines, Initial mapping, 1 × 106 steps of competition, and 3 × 107 steps, respectively. M, Distribution did not change during competition steps. N, Value of truncated peak of the distribution of the initial mapping is 245. O, Moderate solid line, Map on 6 × 104 steps. Value of the truncated peak of the distribution was 1000 of the initial mapping. P, Value of the truncated peak of the distribution of the initial mapping was 583. C, R, Caudal and rostral SC, respectively; N, T, nasal and temporal retina, respectively.
- Figure 8.
Simulations for knock-out mice with upregulation of Eph in the retina attributable to disrupted ephrins. A-D, Graded receptor densities in the retina used for computer simulations. Dotted-dashed line (line w), 1.224 f(xR), the same as line w in A (for comparison). A, Receptor density of wild-type mouse (line w), 1.224 f(xR). B, Solid line, Receptor density of A2-/- (line w + a′), 1.224 f(xR) + 0.45 fa(100 - xR); dotted line (line a′), 0.45 fa(100 - xR). C, Solid line, Receptor density of A5-/- (line w+b′),1.224f(xR)+0.6fb(100-xR);arrow, position of a trough; dotted line (line b′), 0.6 fb (100-xR). D, Solid line, Receptor density of heterozygote A2+/- and A5+/-, line w + (a′+ b′)/2, 1.224 f(xR) + 1/2 [0.6 fa(100 - xR) + 0.45 fb(100 - xR)]. Dotted line, (a′+ b′)/2, 1/2 [0.6 fa(100 - xR) + 0.45 fb(100 - xR)]. E-H, Graded ligand densities in the SC used for computer simulations. Dotted-dashed line, 1.224 f(xL) the same of line w in E (for comparison). E, Solid line (line w), Total ligand density of lines a and b, 1.224 f(xL); dotted line (line a), 1.224 fa(xL); broken line (line b), 1.224 fb(xL). F, Ligand density of A2-/-, 1.224 fb(xL). G, Ligand density of A5-/-, 1.224 fa(xL). H, Solid line, Ligand density of heterozygote A2+/- and A5+/-, line (a + b)/2, 1/2 [1.224 fb(xL) + 1.224 fa(xL)]. In the above-mentioned explanation, the following functions are used: f(x) = 20 exp(x/ 70); fa(x) = 16 exp(x/180); and fb(x) = f(x) - fa(x). I-L, Topographic maps by computer simulations. Thin, gray, thick lines, Initial mapping, 5 × 106 steps of competition, and 2 × 107 steps, respectively; open, solid circles, projected positions of nasal and temporal retinal axon terminals on 2 × 107 steps. I, Map by the ligand density of E (line w) and the receptor density of A (line w). The map did not change during competition steps. J, Map by the ligand density of F (line b) and the receptor density of B (line w + a′). Open circles, xL = 100. K, Map by the ligand density of G (line a) and the receptor density of C (line w + b′). Open circles, xL = 71; solid circle, xL = 21; arrow, xR = 20. L, Map of heterozygote by the ligand density of H [solid line, line (a + b)/2] and the receptor density of figure D [solid line, line w + (a′+ b′)/2]. Open circles, xL = 100; solid circles, xL = 21. M-P, Distribution of population densities of axon terminals on the SC. Thin, gray, thick solid lines, Initial mapping, 5 × 106 steps of competition, and 2 × 107 steps, respectively. M, Distribution did not change during competition steps. N, Distribution changed during competition steps. O, Value of the truncated peak of the distribution of the initial mapping was 703. P, Value of the truncated peak of the distribution of the initial mapping was 222. C, R, Caudal and rostral SC, respectively; N, T, Nasal and temporal retina, respectively.
- Figure 9.
Various combinations of the ligand density gradient on the SC and receptor density gradient on the retina that produce the topographic maps of Figure 1C (dotted lines a, b). Solid line, Wild-type level of receptors or ligands; broken line, elevated level of receptor gradient by ectopic receptor RE. RE was defined so that the simulation result becomes close to the experimental result. C,R, Caudal and rostral SC, respectively; N, T, nasal and temporal retina, respectively. A, Exponential function, R(xR) = S1/2 exp[a(xR-50)] + RE, where a = 1/80, and RE = 36 (broken line) or 0 (solid line). B, Exponential function, L(xL) = S1/2 exp[a (xL - 50)], where a = 1/80. C, Exponential function of moderate slope, R(xR), same as A, but a = 1/160, and RE = 20 (broken line) or 0 (solid line). D, Exponential function of moderate slope, L(xL), same as B, but a = 1/160. E, Hyperbola, R(xR) = S/[0.6(xR - 50) + 50] + RE, where RE = 40 (broken line) or 0 (solid line). F, Linear, L(xL) = 0.6(50 - xL) + 50. G, Linear, R(xR) = 0.6(xR - 50) + 50 + RE, where RE = 40 (broken line) or 0 (solid line). H, Hyperbola, L(xL) = S/[0.6(50 - xL) + 50].
