{
  "slug": "2026-09-09-morphogen-gradient-scaling",
  "exponential_examples": [
    {
      "case": 0,
      "L": 50.0,
      "decay_length": 10.0,
      "amplitude": 1.0,
      "threshold": 0.2,
      "relative_threshold_position": 0.3218875824868201
    },
    {
      "case": 0,
      "L": 100.0,
      "decay_length": 10.0,
      "amplitude": 1.0,
      "threshold": 0.2,
      "relative_threshold_position": 0.16094379124341004
    },
    {
      "case": 1,
      "L": 50.0,
      "decay_length": 10.0,
      "amplitude": 1.0,
      "threshold": 0.2,
      "relative_threshold_position": 0.3218875824868201
    },
    {
      "case": 1,
      "L": 100.0,
      "decay_length": 20.0,
      "amplitude": 1.0,
      "threshold": 0.2,
      "relative_threshold_position": 0.3218875824868201
    },
    {
      "case": 2,
      "L": 50.0,
      "decay_length": 10.0,
      "amplitude": 1.0,
      "threshold": 0.2,
      "relative_threshold_position": 0.3218875824868201
    },
    {
      "case": 2,
      "L": 100.0,
      "decay_length": 20.0,
      "amplitude": 2.0,
      "threshold": 0.2,
      "relative_threshold_position": 0.4605170185988092
    }
  ],
  "source_sink_counterexample": {
    "model": "D M''=0, M(0)=1, M(L)=0",
    "threshold": 0.2,
    "relative_threshold_position": 0.8,
    "condition": "No bulk loss and both boundary concentrations fixed; a special alternative to feedback, not a general tissue mechanism."
  },
  "feedback_parameters": {
    "DM": 1.0,
    "k": 1.0,
    "nuM": 5.0,
    "nuE": 0.02,
    "mu": 0.01,
    "m": 0.5,
    "h": 4,
    "beta": 0.2
  },
  "DE_values": {
    "fast_expander": 10.0,
    "slow_expander": 0.01
  },
  "feedback": {
    "fast_expander": {
      "M_left": [
        68.51611860701222,
        172.51771348017874
      ],
      "E_right": [
        14.414634425641912,
        37.231175077318454
      ],
      "anchors": {
        "0.3": {
          "threshold": 10.093248214227074,
          "r2": 0.33133511405303023,
          "shift": 0.031335114053030244
        },
        "0.5": {
          "threshold": 0.7649586435244276,
          "r2": 0.4876099940502309,
          "shift": -0.012390005949769078
        },
        "0.8": {
          "threshold": 0.016574059282474372,
          "r2": 0.7223452778049458,
          "shift": -0.0776547221950542
        },
        "0.9": {
          "threshold": 0.0049668704734167814,
          "r2": 0.7967052684452652,
          "shift": -0.10329473155473479
        }
      },
      "unmatched_thresholds_in_tested_range": 0
    },
    "slow_expander": {
      "M_left": [
        5.010101194880026,
        5.001001484317123
      ],
      "E_right": [
        346.26486748709056,
        1384.3587221820771
      ],
      "anchors": {
        "0.3": {
          "threshold": 1.6401471047996834,
          "r2": 0.30230878619674906,
          "shift": 0.0023087861967490686
        },
        "0.5": {
          "threshold": 0.6075458696681751,
          "r2": 0.4986152992656765,
          "shift": -0.0013847007343235074
        },
        "0.8": {
          "threshold": 0.3048634361222351,
          "r2": 0.7947906100039027,
          "shift": -0.0052093899960973244
        },
        "0.9": {
          "threshold": 0.2745622293044799,
          "r2": 0.8899156012617592,
          "shift": -0.010084398738240852
        }
      },
      "unmatched_thresholds_in_tested_range": 0
    }
  },
  "max_threshold_position_difference_400_800": 2.0412742757158764e-05,
  "solver_records": [
    "feedback-explore-n400.json",
    "feedback-explore-n800.json"
  ],
  "limits": "Illustrative model parameters, no original Julia random sweep reproduced. Source width scales by assumption. All threshold coordinates use whole-domain x/L, not target-only rescaling. Source-region concentration peaks are retained; monotonic interpolation only used in target tissue. Two steady sizes are not an actual tissue-growth trajectory. No intrinsic-noise or precision simulation.",
  "provenance": "Mosby & Hadjivasiliou (2026) Eqs.2–3; original Julia fKFeedback! checked. Figures and numerical integration independently written in Python."
}